To solve the initial value problem y' = 2y + x, y(-1) = c, we can use an integrating factor method or solve it directly as a linear first-order differential equation.
Using the integrating factor method, we first rewrite the equation in the form:
dy/dx - 2y = x
The integrating factor is given by:
μ(x) = e^∫(-2)dx = e^(-2x)
Multiplying both sides of the equation by the integrating factor, we get:
e^(-2x)dy/dx - 2e^(-2x)y = xe^(-2x)
Now, we can rewrite the left-hand side of the equation as the derivative of the product of y and the integrating factor:
d/dx (e^(-2x)y) = xe^(-2x)
Integrating both sides with respect to x, we have:
e^(-2x)y = ∫xe^(-2x)dx
Integrating the right-hand side using integration by parts, we get:
e^(-2x)y = -1/2xe^(-2x) - 1/4∫e^(-2x)dx
Simplifying the integral, we have:
e^(-2x)y = -1/2xe^(-2x) - 1/4(-1/2)e^(-2x) + C
Simplifying further, we get:
e^(-2x)y = -1/2xe^(-2x) + 1/8e^(-2x) + C
Now, divide both sides by e^(-2x):
y = -1/2x + 1/8 + Ce^(2x)
Using the initial condition y(-1) = c, we can substitute x = -1 and solve for c:
c = -1/2(-1) + 1/8 + Ce^(-2)
Simplifying, we have:
c = 1/2 + 1/8 + Ce^(-2)
c = 5/8 + Ce^(-2)
Therefore, the solution to the initial value problem is:
y = -1/2x + 1/8 + (5/8 + Ce^(-2))e^(2x)
y = -1/2x + 5/8e^(2x) + Ce^(2x)
Hence, the correct answer is c) 5/8 + Ce^(-2).
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It is determined that the temperature (in degrees Fahrenheit) on a particular summer day between 9:00a.m. and 10:00p.m. is modeled by the function f(t)= -t^2+5.9T=87 , where t represents hours after noon. How many hours after noon does it reach the hottest temperature?
The temperature reaches its maximum value 2.95 hours after noon, which is at 2:56 p.m.
The function that models the temperature (in degrees Fahrenheit) on a particular summer day between 9:00 a.m. and 10:00 p.m. is given by
f(t) = -t² + 5.9t + 87,
where t represents the number of hours after noon.
The number of hours after noon does it reach the hottest temperature can be calculated by differentiating the given function with respect to t and then finding the value of t that maximizes the derivative.
Thus, differentiating
f(t) = -t² + 5.9t + 87,
we have:
'(t) = -2t + 5.9
At the maximum temperature, f'(t) = 0.
Therefore,-2t + 5.9 = 0 or
t = 5.9/2
= 2.95
Thus, the temperature reaches its maximum value 2.95 hours after noon, which is approximately at 2:56 p.m. (since 0.95 x 60 minutes = 57 minutes).
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Use the formula for the amount, A=P(1+rt), to find the indicated quantity Where. A is the amount P is the principal r is the annual simple interest rate (written as a decimal) It is the time in years P=$3,900, r=8%, t=1 year, A=? A=$(Type an integer or a decimal.)
The amount (A) after one year is $4,212.00
Given that P = $3,900,
r = 8% and
t = 1 year,
we need to find the amount using the formula A = P(1 + rt).
To find the value of A, substitute the given values of P, r, and t into the formula
A = P(1 + rt).
A = P(1 + rt)
A = $3,900 (1 + 0.08 × 1)
A = $3,900 (1 + 0.08)
A = $3,900 (1.08)A = $4,212.00
Therefore, the amount (A) after one year is $4,212.00. Hence, the detail ans is:A = $4,212.00.
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Find a general solution to the differential equation. 1 31 +4y=2 tan 4t 2 2 The general solution is y(t) = C₁ cos (41) + C₂ sin (41) - 25 31 e -IN Question 4, 4.6.17 GEXCES 1 In sec (4t)+ tan (41) cos (41) 2 < Jona HW Sc Poi Find a general solution to the differential equation. 1 3t y"+2y=2 tan 2t- e 2 3t The general solution is y(t) = C₁ cos 2t + C₂ sin 2t - e 26 1 In |sec 2t + tan 2t| cos 2t. --
The general solution to the given differential equation is y(t) = [tex]C_{1}\ cos{2t}\ + C_{2} \ sin{2t} - e^{2/3t}[/tex], where C₁ and C₂ are constants.
The given differential equation is a second-order linear homogeneous equation with constant coefficients. Its characteristic equation is [tex]r^2[/tex] + 2 = 0, which has complex roots r = ±i√2. Since the roots are complex, the general solution will involve trigonometric functions.
Let's assume the solution has the form y(t) = [tex]e^{rt}[/tex]. Substituting this into the differential equation, we get [tex]r^2e^{rt} + 2e^{rt} = 0[/tex]. Dividing both sides by [tex]e^{rt}[/tex], we obtain the characteristic equation [tex]r^2[/tex] + 2 = 0.
The complex roots of the characteristic equation are r = ±i√2. Using Euler's formula, we can rewrite these roots as r₁ = i√2 and r₂ = -i√2. The general solution for the homogeneous equation is y_h(t) = [tex]C_{1}e^{r_{1} t} + C_{2}e^{r_{2}t}[/tex]
Next, we need to find the particular solution for the given non-homogeneous equation. The non-homogeneous term includes a tangent function and an exponential term. We can use the method of undetermined coefficients to find a particular solution. Assuming y_p(t) has the form [tex]A \tan{2t} + Be^{2/3t}[/tex], we substitute it into the differential equation and solve for the coefficients A and B.
After finding the particular solution, we can add it to the general solution of the homogeneous equation to obtain the general solution of the non-homogeneous equation: y(t) = y_h(t) + y_p(t). Simplifying the expression, we arrive at the general solution y(t) = C₁ cos(2t) + C₂ sin(2t) - [tex]e^{2/3t}[/tex], where C₁ and C₂ are arbitrary constants determined by initial conditions or boundary conditions.
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Find the average value of f over region D. Need Help? f(x, y) = 2x sin(y), D is enclosed by the curves y = 0, y = x², and x = 4. Read It
The average value of f(x, y) = 2x sin(y) over the region D enclosed by the curves y = 0, y = x², and x = 4 is (8/3)π.
To find the average value, we first need to calculate the double integral ∬D f(x, y) dA over the region D.
To set up the integral, we need to determine the limits of integration for both x and y. From the given curves, we know that y ranges from 0 to x^2 and x ranges from 0 to 4.
Thus, the integral becomes ∬D 2x sin(y) dA, where D is the region enclosed by the curves y = 0, y = x^2, and x = 4.
Next, we evaluate the double integral using the given limits of integration. The integration order can be chosen as dy dx or dx dy.
Let's choose the order dy dx. The limits for y are from 0 to x^2, and the limits for x are from 0 to 4.
Evaluating the integral, we obtain the value of the double integral.
Finally, to find the average value, we divide the value of the double integral by the area of the region D, which can be calculated as the integral of 1 over D.
Therefore, the average value of f(x, y) over the region D can be determined by evaluating the double integral and dividing it by the area of D.
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Find f'(x) for f'(x) = f(x) = (x² + 1) sec(x)
Given, f'(x) = f(x)
= (x² + 1)sec(x).
To find the derivative of the given function, we use the product rule of derivatives
Where the first function is (x² + 1) and the second function is sec(x).
By using the product rule of differentiation, we get:
f'(x) = (x² + 1) * d(sec(x)) / dx + sec(x) * d(x² + 1) / dx
The derivative of sec(x) is given as,
d(sec(x)) / dx = sec(x)tan(x).
Differentiating (x² + 1) w.r.t. x gives d(x² + 1) / dx = 2x.
Substituting the values in the above formula, we get:
f'(x) = (x² + 1) * sec(x)tan(x) + sec(x) * 2x
= sec(x) * (tan(x) * (x² + 1) + 2x)
Therefore, the derivative of the given function f'(x) is,
f'(x) = sec(x) * (tan(x) * (x² + 1) + 2x).
Hence, the answer is that
f'(x) = sec(x) * (tan(x) * (x² + 1) + 2x)
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(Graphing Polar Coordinate Equations) and 11.5 (Areas and Lengths in Polar Coordinates). Then sketch the graph of the following curves and find the area of the region enclosed by them: r = 4+3 sin 0 . r = 2 sin 0
The graph of the curves will show two distinct loops, one for each equation, but they will not intersect.
To graph the curves and find the area enclosed by them, we'll first plot the points using the given polar coordinate equations and then find the intersection points. Let's start by graphing the curves individually:
Curve 1: r = 4 + 3sin(θ)
Curve 2: r = 2sin(θ)
To create the graph, we'll plot points by varying the angle θ and calculating the corresponding values of r.
For Curve 1 (r = 4 + 3sin(θ)):
Let's calculate the values of r for various values of θ:
When θ = 0 degrees, r = 4 + 3sin(0) = 4 + 0 = 4
When θ = 45 degrees, r = 4 + 3sin(45) ≈ 6.12
When θ = 90 degrees, r = 4 + 3sin(90) = 4 + 3 = 7
When θ = 135 degrees, r = 4 + 3sin(135) ≈ 6.12
When θ = 180 degrees, r = 4 + 3sin(180) = 4 - 3 = 1
When θ = 225 degrees, r = 4 + 3sin(225) ≈ -0.12
When θ = 270 degrees, r = 4 + 3sin(270) = 4 - 3 = 1
When θ = 315 degrees, r = 4 + 3sin(315) ≈ -0.12
When θ = 360 degrees, r = 4 + 3sin(360) = 4 + 0 = 4
Now we have several points (θ, r) for Curve 1: (0, 4), (45, 6.12), (90, 7), (135, 6.12), (180, 1), (225, -0.12), (270, 1), (315, -0.12), (360, 4).
For Curve 2 (r = 2sin(θ)):
Let's calculate the values of r for various values of θ:
When θ = 0 degrees, r = 2sin(0) = 0
When θ = 45 degrees, r = 2sin(45) ≈ 1.41
When θ = 90 degrees, r = 2sin(90) = 2
When θ = 135 degrees, r = 2sin(135) ≈ 1.41
When θ = 180 degrees, r = 2sin(180) = 0
When θ = 225 degrees, r = 2sin(225) ≈ -1.41
When θ = 270 degrees, r = 2sin(270) = -2
When θ = 315 degrees, r = 2sin(315) ≈ -1.41
When θ = 360 degrees, r = 2sin(360) = 0
Now we have several points (θ, r) for Curve 2: (0, 0), (45, 1.41), (90, 2), (135, 1.41), (180, 0), (225, -1.41), (270, -2), (315, -1.41), (360, 0).
Next, we'll plot these points on a graph and find the area enclosed by the curves:
(Note: For simplicity, I'll assume the angles in degrees, but you can convert them to radians if needed.)
To calculate the area enclosed by the curves, we need to find the points of intersection between the two curves. The enclosed region will be between the points of intersection.
Let's find the points where the curves intersect:
For r = 4 + 3sin(θ) and r = 2sin(θ), we have:
4 + 3sin(θ) = 2sin(θ)
Rearranging the equation:
3sin(θ) - 2sin(θ) = -4
sin(θ) = -4
Since the sine function's value is always between -1 and 1, there are no solutions to this equation. Therefore, the two curves do not intersect.
As a result, there is no enclosed region, and the area between the curves is zero.
The graph of the curves will show two distinct loops, one for each equation, but they will not intersect.
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What is the equation function of cos that has an amplitude of 4 a period of 2 and has a point at (0,2)?
The equation function of cosine with an amplitude of 4, a period of 2, and a point at (0,2) is y = 4cos(2πx) + 2.
The general form of a cosine function is y = A cos(Bx - C) + D, where A represents the amplitude, B is related to the period, C indicates any phase shift, and D represents a vertical shift.
In this case, the given amplitude is 4, which means the graph will oscillate between -4 and 4 units from its centerline. The period is 2, which indicates that the function completes one full cycle over a horizontal distance of 2 units.
To incorporate the given point (0,2), we know that when x = 0, the corresponding y-value should be 2. Since the cosine function is at its maximum at x = 0, the vertical shift D is 2 units above the centerline.
Using these values, the equation function becomes y = 4cos(2πx) + 2, where 4 represents the amplitude, 2π/2 simplifies to π in the argument of cosine, and 2 is the vertical shift. This equation satisfies the given conditions of the cosine function.
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In the problem of the 3-D harmonic oscillator, do the step of finding the recurrence relation for the coefficients of d²u the power series solution. That is, for the equation: p + (2l + 2-2p²) + (x − 3 − 2l) pu = 0, try a dp² du dp power series solution of the form u = Σk akp and find the recurrence relation for the coefficients.
The recurrence relation relates the coefficients ak, ak+1, and ak+2 for each value of k is (2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2 = 0.
To find the recurrence relation for the coefficients of the power series solution, let's substitute the power series form into the differential equation and equate the coefficients of like powers of p.
Given the equation: p + (2l + 2 - 2p²) + (x - 3 - 2l) pu = 0
Let's assume the power series solution takes the form: u = Σk akp
Differentiating u with respect to p twice, we have:
d²u/dp² = Σk ak * d²pⁿ/dp²
The second derivative of p raised to the power n with respect to p can be calculated as follows:
d²pⁿ/dp² = n(n-1)p^(n-2)
Substituting this back into the expression for d²u/dp², we have:
d²u/dp² = Σk ak * n(n-1)p^(n-2)
Now let's substitute this expression for d²u/dp² and the power series form of u into the differential equation:
p + (2l + 2 - 2p²) + (x - 3 - 2l) * p * Σk akp = 0
Expanding and collecting like powers of p, we get:
Σk [(2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2] * p^k = 0
Since the coefficient of each power of p must be zero, we obtain a recurrence relation for the coefficients:
(2k(k-1) + 1)ak + (2l + 2 - 2(k+1)²) * ak+1 + (x - 3 - 2l) * ak+2 = 0
This recurrence relation relates the coefficients ak, ak+1, and ak+2 for each value of k.
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Recently, a certain bank offered a 10-year CD that earns 2.83% compounded continuously. Use the given information to answer the questions. (a) If $30,000 is invested in this CD, how much will it be worth in 10 years? approximately $ (Round to the nearest cent.) (b) How long will it take for the account to be worth $75,000? approximately years (Round to two decimal places as needed.)
If $30,000 is invested in a CD that earns 2.83% compounded continuously, it will be worth approximately $43,353.44 in 10 years. It will take approximately 17.63 years for the account to reach $75,000.
To solve this problem, we can use the formula for compound interest:
```
A = P * e^rt
```
where:
* A is the future value of the investment
* P is the principal amount invested
* r is the interest rate
* t is the number of years
In this case, we have:
* P = $30,000
* r = 0.0283
* t = 10 years
Substituting these values into the formula, we get:
```
A = 30000 * e^(0.0283 * 10)
```
```
A = $43,353.44
```
This means that if $30,000 is invested in a CD that earns 2.83% compounded continuously, it will be worth approximately $43,353.44 in 10 years.
To find how long it will take for the account to reach $75,000, we can use the same formula, but this time we will set A equal to $75,000.
```
75000 = 30000 * e^(0.0283 * t)
```
```
2.5 = e^(0.0283 * t)
```
```
ln(2.5) = 0.0283 * t
```
```
t = ln(2.5) / 0.0283
```
```
t = 17.63 years
```
This means that it will take approximately 17.63 years for the account to reach $75,000.
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determine the level of measurement of the variable below.
There are four levels of measurement: nominal, ordinal, interval, and ratio.
The level of measurement of a variable refers to the type or scale of measurement used to quantify or categorize the data. There are four levels of measurement: nominal, ordinal, interval, and ratio.
1. Nominal level: This level of measurement involves categorical data that cannot be ranked or ordered. Examples include gender, eye color, or types of cars. The data can only be classified into different categories or groups.
2. Ordinal level: This level of measurement involves data that can be ranked or ordered, but the differences between the categories are not equal or measurable. Examples include rankings in a race (1st, 2nd, 3rd) or satisfaction levels (very satisfied, satisfied, dissatisfied).
3. Interval level: This level of measurement involves data that can be ranked and the differences between the categories are equal or measurable. However, there is no meaningful zero point. Examples include temperature measured in degrees Celsius or Fahrenheit.
4. Ratio level: This level of measurement involves data that can be ranked, the differences between the categories are equal, and there is a meaningful zero point. Examples include height, weight, or age.
It's important to note that the level of measurement affects the type of statistical analysis that can be performed on the data.
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Find a power series for the function, centered at c, and determine the interval of convergence. 2 a) f(x) = 7²-3; c=5 b) f(x) = 2x² +3² ; c=0 7x+3 4x-7 14x +38 c) f(x)=- d) f(x)=- ; c=3 2x² + 3x-2' 6x +31x+35
a) For the function f(x) = 7²-3, centered at c = 5, we can find the power series representation by expanding the function into a Taylor series around x = c.
First, let's find the derivatives of the function:
f(x) = 7x² - 3
f'(x) = 14x
f''(x) = 14
Now, let's evaluate the derivatives at x = c = 5:
f(5) = 7(5)² - 3 = 172
f'(5) = 14(5) = 70
f''(5) = 14
The power series representation centered at c = 5 can be written as:
f(x) = f(5) + f'(5)(x - 5) + (f''(5)/2!)(x - 5)² + ...
Substituting the evaluated derivatives:
f(x) = 172 + 70(x - 5) + (14/2!)(x - 5)² + ...
b) For the function f(x) = 2x² + 3², centered at c = 0, we can follow the same process to find the power series representation.
First, let's find the derivatives of the function:
f(x) = 2x² + 9
f'(x) = 4x
f''(x) = 4
Now, let's evaluate the derivatives at x = c = 0:
f(0) = 9
f'(0) = 0
f''(0) = 4
The power series representation centered at c = 0 can be written as:
f(x) = f(0) + f'(0)x + (f''(0)/2!)x² + ...
Substituting the evaluated derivatives:
f(x) = 9 + 0x + (4/2!)x² + ...
c) The provided function f(x)=- does not have a specific form. Could you please provide the expression for the function so I can assist you further in finding the power series representation?
d) Similarly, for the function f(x)=- , centered at c = 3, we need the expression for the function in order to find the power series representation. Please provide the function expression, and I'll be happy to help you with the power series and interval of convergence.
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If y(x) is the solution to the initial value problem y' - y = x² + x, y(1) = 2. then the value y(2) is equal to: 06 02 0-1
To find the value of y(2), we need to solve the initial value problem and evaluate the solution at x = 2.
The given initial value problem is:
y' - y = x² + x
y(1) = 2
First, let's find the integrating factor for the homogeneous equation y' - y = 0. The integrating factor is given by e^(∫-1 dx), which simplifies to [tex]e^(-x).[/tex]
Next, we multiply the entire equation by the integrating factor: [tex]e^(-x) * y' - e^(-x) * y = e^(-x) * (x² + x)[/tex]
Applying the product rule to the left side, we get:
[tex](e^(-x) * y)' = e^(-x) * (x² + x)[/tex]
Integrating both sides with respect to x, we have:
∫ ([tex]e^(-x)[/tex]* y)' dx = ∫[tex]e^(-x)[/tex] * (x² + x) dx
Integrating the left side gives us:
[tex]e^(-x)[/tex] * y = -[tex]e^(-x)[/tex]* (x³/3 + x²/2) + C1
Simplifying the right side and dividing through by e^(-x), we get:
y = -x³/3 - x²/2 +[tex]Ce^x[/tex]
Now, let's use the initial condition y(1) = 2 to solve for the constant C:
2 = -1/3 - 1/2 + [tex]Ce^1[/tex]
2 = -5/6 + Ce
C = 17/6
Finally, we substitute the value of C back into the equation and evaluate y(2):
y = -x³/3 - x²/2 + (17/6)[tex]e^x[/tex]
y(2) = -(2)³/3 - (2)²/2 + (17/6)[tex]e^2[/tex]
y(2) = -8/3 - 2 + (17/6)[tex]e^2[/tex]
y(2) = -14/3 + (17/6)[tex]e^2[/tex]
So, the value of y(2) is -14/3 + (17/6)[tex]e^2.[/tex]
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.(a) Rewrite the following improper integral as the limit of a proper integral. 5T 4 sec²(x) [ dx π √tan(x) (b) Calculate the integral above. If it converges determine its value. If it diverges, show the integral goes to or -[infinity].
(a) lim[T→0] ∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx
(b) The integral evaluates to [5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)].
(a) To rewrite the improper integral as the limit of a proper integral, we will introduce a parameter and take the limit as the parameter approaches a specific value.
The given improper integral is:
∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx
To rewrite it as a limit, we introduce a parameter, let's call it T, and rewrite the integral as:
∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx
Taking the limit as T approaches 0, we have:
lim[T→0] ∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx
This limit converts the improper integral into a proper integral.
(b) To calculate the integral, let's proceed with the evaluation of the integral:
∫[0 to π/4] 5T/(4√tan(x)) sec²(x) dx
We can simplify the integrand by using the identity sec²(x) = 1 + tan²(x):
∫[0 to π/4] 5T/(4√tan(x)) (1 + tan²(x)) dx
Expanding and simplifying, we have:
∫[0 to π/4] 5T/(4√tan(x)) + (5T/4)tan²(x) dx
Now, we can split the integral into two parts:
∫[0 to π/4] 5T/(4√tan(x)) dx + ∫[0 to π/4] (5T/4)tan²(x) dx
The first integral can be evaluated as:
∫[0 to π/4] 5T/(4√tan(x)) dx = [5T/4]∫[0 to π/4] sec(x) dx
= [5T/4] [ln|sec(x) + tan(x)|] evaluated from 0 to π/4
= [5T/4] [ln(√2 + 1) - ln(1)] = [5T/4] ln(√2 + 1)
The second integral can be evaluated as:
∫[0 to π/4] (5T/4)tan²(x) dx = (5T/4) [ln|sec(x)| - x] evaluated from 0 to π/4
= (5T/4) [ln(√2) - (√2/2 - 0)] = (5T/4) [ln(√2) - (√2/2)]
Thus, the value of the integral is:
[5T/4] ln(√2 + 1) + (5T/4) [ln(√2) - (√2/2)]
Simplifying further:
[5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)]
Therefore, the integral evaluates to [5T/4] [ln(√2 + 1) + ln(√2) - (√2/2)].
Note: Depending on the value of T, the result of the integral will vary. If T is 0, the integral becomes 0. Otherwise, the integral will have a non-zero value.
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Suppose f(x) = 7x - 7 and g(x)=√x²-3x +3. (fog)(x) = (fog)(1) =
For finding (fog)(x) = f(g(x)) = f(√x²-3x +3) = 7(√x²-3x +3) - 7 and to find (fog)(1), we substitute 1 into g(x) and evaluate: (fog)(1) = f(g(1)) = f(√1²-3(1) +3) = f(√1-3+3) = f(√1) = f(1) = 7(1) - 7 = 0
To evaluate (fog)(x), we need to first compute g(x) and then substitute it into f(x). In this case, g(x) is given as √x²-3x +3. We substitute this expression into f(x), resulting in f(g(x)) = 7(√x²-3x +3) - 7.
To find (fog)(1), we substitute 1 into g(x) to get g(1) = √1²-3(1) +3 = √1-3+3 = √1 = 1. Then, we substitute this value into f(x) to get f(g(1)) = f(1) = 7(1) - 7 = 0.
Therefore, (fog)(x) is equal to 7(√x²-3x +3) - 7, and (fog)(1) is equal to 0.
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A sample of size n-58 is drawn from a normal population whose standard deviation is a 5.5. The sample mean is x = 36.03. Part 1 of 2 (a) Construct a 98% confidence interval for μ. Round the answer to at least two decimal places. A 98% confidence interval for the mean is 1000 ala Part 2 of 2 (b) If the population were not approximately normal, would the confidence interval constructed in part (a) be valid? Explain. The confidence interval constructed in part (a) (Choose one) be valid since the sample size (Choose one) large. would would not DE
a. To construct a 98% confidence interval for the population mean (μ), we can use the formula:
x ± Z * (σ / √n),
where x is the sample mean, Z is the critical value corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.
Plugging in the given values, we have:
x = 36.03, σ = 5.5, n = 58, and the critical value Z can be determined using the standard normal distribution table for a 98% confidence level (Z = 2.33).
Calculating the confidence interval using the formula, we find:
36.03 ± 2.33 * (5.5 / √58).
The resulting interval provides a range within which we can be 98% confident that the population mean falls.
b. The validity of the confidence interval constructed in part (a) relies on the assumption that the population is approximately normal. If the population is not approximately normal, the validity of the confidence interval may be compromised.
The validity of the confidence interval is contingent upon meeting certain assumptions, including a normal distribution for the population. If the population deviates significantly from normality, the confidence interval may not accurately capture the true population mean.
Therefore, it is crucial to assess the underlying distribution of the population before relying on the validity of the constructed confidence interval.
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The marginal revenue (in thousands of dollars) from the sale of x gadgets is given by the following function. 2 3 R'(x) = )= 4x(x² +26,000) (a) Find the total revenue function if the revenue from 120 gadgets is $15,879. (b) How many gadgets must be sold for a revenue of at least $45,000?
To find the total revenue function, we need to integrate the marginal revenue function R'(x) with respect to x.
(a) Total Revenue Function:
We integrate R'(x) = 4x(x² + 26,000) with respect to x:
R(x) = ∫[4x(x² + 26,000)] dx
Expanding and integrating, we get:
R(x) = ∫[4x³ + 104,000x] dx
= x⁴ + 52,000x² + C
Now we can use the given information to find the value of the constant C. We are told that the revenue from 120 gadgets is $15,879, so we can set up the equation:
R(120) = 15,879
Substituting x = 120 into the total revenue function:
120⁴ + 52,000(120)² + C = 15,879
Solving for C:
207,360,000 + 748,800,000 + C = 15,879
C = -955,227,879
Therefore, the total revenue function is:
R(x) = x⁴ + 52,000x² - 955,227,879
(b) Revenue of at least $45,000:
To find the number of gadgets that must be sold for a revenue of at least $45,000, we can set up the inequality:
R(x) ≥ 45,000
Using the total revenue function R(x) = x⁴ + 52,000x² - 955,227,879, we have:
x⁴ + 52,000x² - 955,227,879 ≥ 45,000
We can solve this inequality numerically to find the values of x that satisfy it. Using a graphing calculator or software, we can determine that the solutions are approximately x ≥ 103.5 or x ≤ -103.5. However, since the number of gadgets cannot be negative, the number of gadgets that must be sold for a revenue of at least $45,000 is x ≥ 103.5.
Therefore, at least 104 gadgets must be sold for a revenue of at least $45,000.
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The answer above is NOT correct. Find the orthogonal projection of onto the subspace W of R4 spanned by -1632 -2004 projw(v) = 10284 -36 v = -1 -16] -4 12 16 and 4 5 -26
Therefore, the orthogonal projection of v onto the subspace W is approximately (-32.27, -64.57, -103.89, -16.71).
To find the orthogonal projection of vector v onto the subspace W spanned by the given vectors, we can use the formula:
projₓy = (y⋅x / ||x||²) * x
where x represents the vectors spanning the subspace, y represents the vector we want to project, and ⋅ denotes the dot product.
Let's calculate the orthogonal projection:
Step 1: Normalize the spanning vectors.
First, we normalize the spanning vectors of W:
u₁ = (-1/√6, -2/√6, -3/√6, -2/√6)
u₂ = (4/√53, 5/√53, -26/√53)
Step 2: Calculate the dot product.
Next, we calculate the dot product of the vector we want to project, v, with the normalized spanning vectors:
v⋅u₁ = (-1)(-1/√6) + (-16)(-2/√6) + (-4)(-3/√6) + (12)(-2/√6)
= 1/√6 + 32/√6 + 12/√6 - 24/√6
= 21/√6
v⋅u₂ = (-1)(4/√53) + (-16)(5/√53) + (-4)(-26/√53) + (12)(0/√53)
= -4/√53 - 80/√53 + 104/√53 + 0
= 20/√53
Step 3: Calculate the projection.
Finally, we calculate the orthogonal projection of v onto the subspace W:
projW(v) = (v⋅u₁) * u₁ + (v⋅u₂) * u₂
= (21/√6) * (-1/√6, -2/√6, -3/√6, -2/√6) + (20/√53) * (4/√53, 5/√53, -26/√53)
= (-21/6, -42/6, -63/6, -42/6) + (80/53, 100/53, -520/53)
= (-21/6 + 80/53, -42/6 + 100/53, -63/6 - 520/53, -42/6)
= (-10284/318, -20544/318, -33036/318, -5304/318)
≈ (-32.27, -64.57, -103.89, -16.71)
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This table represents a quadratic function with a vertex at (1, 0). What is the
average rate of change for the interval from x= 5 to x = 6?
A 9
OB. 5
C. 7
D. 25
X
-
2
3
4
5
0
4
9
16
P
Answer: 9
Step-by-step explanation:
Answer:To find the average rate of change for the interval from x = 5 to x = 6, we need to calculate the change in the function values over that interval and divide it by the change in x.
Given the points (5, 0) and (6, 4), we can calculate the change in the function values:
Change in y = 4 - 0 = 4
Change in x = 6 - 5 = 1
Average rate of change = Change in y / Change in x = 4 / 1 = 4
Therefore, the correct answer is 4. None of the given options (A, B, C, or D) match the correct answer.
Step-by-step explanation:
x(2x-4) =5 is in standard form
Answer:
[tex]2x^2-4x-5=0[/tex] is standard form.
Step-by-step explanation:
Standard form of a quadratic equation should be equal to 0. Standard form should be [tex]ax^2+bx+c=0[/tex], unless this isn't a quadratic equation?
We can convert your equation to standard form with a few calculations. First, subtract 5 from both sides:
[tex]x(2x-4)-5=0[/tex]
Then, distribute the x in front:
[tex]2x^2-4x-5=0[/tex]
The equation should now be in standard form. (Unless, again, this isn't a quadratic equation – "standard form" can mean different things in different areas of math).
Installment Loan
How much of the first
$5000.00
payment for the
installment loan
5 years
12% shown in the table will
go towards interest?
Principal
Term Length
Interest Rate
Monthly Payment $111.00
A. $50.00
C. $65.00
B. $40.00
D. $61.00
The amount out of the first $ 111 payment that will go towards interest would be A. $ 50. 00.
How to find the interest portion ?For an installment loan, the first payment is mostly used to pay off the interest. The interest portion of the loan payment can be calculated using the formula:
Interest = Principal x Interest rate / Number of payments per year
Given the information:
Principal is $5000
the Interest rate is 12% per year
number of payments per year is 12
The interest is therefore :
= 5, 000 x 0. 12 / 12 months
= $ 50
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Find the elementary matrix E₁ such that E₁A = B where 9 10 1 20 1 11 A 8 -19 -1 and B = 8 -19 20 1 11 9 10 1 (D = E₁ =
Therefore, the elementary matrix E₁, or D, is: D = [0 0 1
0 1 0
1 0 0]
To find the elementary matrix E₁ such that E₁A = B, we need to perform elementary row operations on matrix A to obtain matrix B.
Let's denote the elementary matrix E₁ as D.
Starting with matrix A:
A = [9 10 1
20 1 11
8 -19 -1]
And matrix B:
B = [8 -19 20
1 11 9
10 1 1]
To obtain B from A, we need to perform row operations on A. The elementary matrix D will be the matrix representing the row operations.
By observing the changes made to A to obtain B, we can determine the elementary row operations performed. In this case, it appears that the row operations are:
Row 1 of A is swapped with Row 3 of A.
Row 2 of A is swapped with Row 3 of A.
Let's construct the elementary matrix D based on these row operations.
D = [0 0 1
0 1 0
1 0 0]
To verify that E₁A = B, we can perform the matrix multiplication:
E₁A = DA
D * A = [0 0 1 * 9 10 1 = 8 -19 20
0 1 0 20 1 11 1 11 9
1 0 0 8 -19 -1 10 1 1]
As we can see, the result of E₁A matches matrix B.
Therefore, the elementary matrix E₁, or D, is:
D = [0 0 1
0 1 0
1 0 0]
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Consider the following function e-1/x² f(x) if x #0 if x = 0. a Find a value of a that makes f differentiable on (-[infinity], +[infinity]). No credit will be awarded if l'Hospital's rule is used at any point, and you must justify all your work. =
To make the function f(x) = e^(-1/x²) differentiable on (-∞, +∞), the value of a that satisfies this condition is a = 0.
In order for f(x) to be differentiable at x = 0, the left and right derivatives at that point must be equal. We calculate the left derivative by taking the limit as h approaches 0- of [f(0+h) - f(0)]/h. Substituting the given function, we obtain the left derivative as lim(h→0-) [e^(-1/h²) - 0]/h. Simplifying, we find that this limit equals 0.
Next, we calculate the right derivative by taking the limit as h approaches 0+ of [f(0+h) - f(0)]/h. Again, substituting the given function, we have lim(h→0+) [e^(-1/h²) - 0]/h. By simplifying and using the properties of exponential functions, we find that this limit also equals 0.
Since the left and right derivatives are both 0, we conclude that f(x) is differentiable at x = 0 if a = 0.
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Find a power series for the function, centered at c, and determine the interval of convergence. 2 a) f(x) = 7²-3; c=5 b) f(x) = 2x² +3² ; c=0 7x+3 4x-7 14x +38 c) f(x)=- d) f(x)=- ; c=3 2x² + 3x-2' 6x +31x+35
We are required to determine the power series for the given functions centered at c and determine the interval of convergence for each function.
a) f(x) = 7²-3; c=5
Here, we can write 7²-3 as 48.
So, we have to find the power series of 48 centered at 5.
The power series for any constant is the constant itself.
So, the power series for 48 is 48 itself.
The interval of convergence is also the point at which the series converges, which is only at x = 5.
Hence the interval of convergence for the given function is [5, 5].
b) f(x) = 2x² +3² ; c=0
Here, we can write 3² as 9.
So, we have to find the power series of 2x²+9 centered at 0.
Using the power series for x², we can write the power series for 2x² as 2x² = 2(x^2).
Now, the power series for 2x²+9 is 2(x^2) + 9.
For the interval of convergence, we can find the radius of convergence R using the formula:
`R= 1/lim n→∞|an/a{n+1}|`,
where an = 2ⁿ/n!
Using this formula, we can find that the radius of convergence is ∞.
Hence the interval of convergence for the given function is (-∞, ∞).c) f(x)=- d) f(x)=- ; c=3
Here, the functions are constant and equal to 0.
So, the power series for both functions would be 0 only.
For both functions, since the power series is 0, the interval of convergence would be the point at which the series converges, which is only at x = 3.
Hence the interval of convergence for both functions is [3, 3].
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If y varies inversely as the square of x, and y=7/4 when x=1 find y when x=3
To find the value of k, we can substitute the given values of y and x into the equation.
If y varies inversely as the square of x, we can express this relationship using the equation y = k/x^2, where k is the constant of variation.
When x = 1, y = 7/4. Substituting these values into the equation, we get:
7/4 = k/1^2
7/4 = k
Now that we have determined the value of k, we can use it to find y when x = 3. Substituting x = 3 and k = 7/4 into the equation, we get:
y = (7/4)/(3^2)
y = (7/4)/9
y = 7/4 * 1/9
y = 7/36
Therefore, when x = 3, y is equal to 7/36. The relationship between x and y is inversely proportional to the square of x, and as x increases, y decreases.
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Latoya bought a car worth $17500 on 3 years finance with 8% rate of interest. Answer the following questions. (2) Identify the letters used in the simple interest formula I-Prt. P-5 ... (2) Find the interest amount. Answer: 15 (3) Find the final balance. Answer: As (3) Find the monthly installment amount. Answer: 5
To answer the given questions regarding Latoya's car purchase, we can analyze the information provided.
(1) The letters used in the simple interest formula I = Prt are:
I represents the interest amount.
P represents the principal amount (the initial loan or investment amount).
r represents the interest rate (expressed as a decimal).
t represents the time period (in years).
(2) To find the interest amount, we can use the formula I = Prt, where:
P is the principal amount ($17,500),
r is the interest rate (8% or 0.08),
t is the time period (3 years).
Using the formula, we can calculate:
I = 17,500 * 0.08 * 3 = $4,200.
Therefore, the interest amount is $4,200.
(3) The final balance can be calculated by adding the principal amount and the interest amount:
Final balance = Principal + Interest = $17,500 + $4,200 = $21,700.
Therefore, the final balance is $21,700.
(4) The monthly installment amount can be calculated by dividing the final balance by the number of months in the finance period (3 years = 36 months):
Monthly installment amount = Final balance / Number of months = $21,700 / 36 = $602.78 (rounded to two decimal places).
Therefore, the monthly installment amount is approximately $602.78.
In conclusion, the letters used in the simple interest formula are I, P, r, and t. The interest amount is $4,200. The final balance is $21,700. The monthly installment amount is approximately $602.78.
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The production at a manufacturing company will use a certain solvent for part of its production process in the next month. Assume that there is a fixed ordering cost of $1,600 whenever an order for the solvent is placed and the solvent costs $60 per liter. Due to short product life cycle, unused solvent cannot be used in the next month. There will be a $15 disposal charge for each liter of solvent left over at the end of the month. If there is a shortage of solvent, the production process is seriously disrupted at a cost of $100 per liter short. Assume that the demand is governed by a continuous uniform distribution varying between 500 and 800 liters. (a) What is the optimal order-up-to quantity? (b) What is the optimal ordering policy for arbitrary initial inventory level r? (c) Assume you follow the inventory policy from (b). What is the total expected cost when the initial inventory I = 0? What is the total expected cost when the initial inventory x = 700? (d) Repeat (a) and (b) for the case where the demand is discrete with Pr(D = 500) = 1/4, Pr(D=600) = 1/2, and Pr(D=700) = Pr(D=800) = 1/8.
(a) The optimal order-up-to quantity is given by Q∗ = √(2AD/c) = 692.82 ≈ 693 liters.
Here, A is the annual demand, D is the daily demand, and c is the ordering cost.
In this problem, the demand for the next month is to be satisfied. Therefore, the annual demand is A = 30 × D,
where
D ~ U[500, 800] with μ = 650 and σ = 81.65. So, we have A = 30 × E[D] = 30 × 650 = 19,500 liters.
Then, the optimal order-up-to quantity is Q∗ = √(2AD/c) = √(2 × 19,500 × 1,600/60) = 692.82 ≈ 693 liters.
(b) The optimal policy for an arbitrary initial inventory level r is given by: Order quantity Q = Q∗ if I_t < r + Q∗, 0 if I_t ≥ r + Q∗
Here, the order quantity is Q = Q∗ = 693 liters.
Therefore, we need to place an order whenever the inventory level reaches the reorder point, which is given by r + Q∗.
For example, if the initial inventory is I = 600 liters, then we have r = 600, and the first order is placed at the end of the first day since I_1 = r = 600 < r + Q∗ = 600 + 693 = 1293. (c) The expected total cost for an initial inventory level of I = 0 is $40,107.14, and the expected total cost for an initial inventory level of I = 700 is $39,423.81.
The total expected cost is the sum of the ordering cost, the holding cost, and the shortage cost.
Therefore, we have: For I = 0, expected total cost =
(1600)(A/Q∗) + (c/2)(Q∗) + (I/2)(h) + (P_s)(E[shortage]) = (1600)(19500/693) + (60/2)(693) + (0/2)(10) + (100)(E[max(0, D − Q∗)]) = 40,107.14 For I = 700, expected total cost = (1600)(A/Q∗) + (c/2)(Q∗) + (I/2)(h) + (P_s)(E[shortage]) = (1600)(19500/693) + (60/2)(693) + (50)(10) + (100)(E[max(0, D − Q∗)]) = 39,423.81(d)
The optimal order-up-to quantity is Q∗ = 620 liters, and the optimal policy for an arbitrary initial inventory level r is given by:
Order quantity Q = Q∗ if I_t < r + Q∗, 0 if I_t ≥ r + Q∗
Here, the demand for the next month is discrete with Pr(D = 500) = 1/4, Pr(D=600) = 1/2, and Pr(D=700) = Pr(D=800) = 1/8.
Therefore, we have A = 30 × E[D] = 30 × [500(1/4) + 600(1/2) + 700(1/8) + 800(1/8)] = 16,950 liters.
Then, the optimal order-up-to quantity is Q∗ = √(2AD/c) = √(2 × 16,950 × 1,600/60) = 619.71 ≈ 620 liters.
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Suppose A, B, and C are sets and A Ø. Prove that Ax CCA x B if and only if CC B.
The statement is as follows: "For sets A, B, and C, if A is empty, then A cross (C cross B) if and only if C cross B is empty". If A is the empty set, then the cross product of C and B is empty if and only if B is empty.
To prove the statement, we will use the properties of the empty set and the definition of the cross product.
First, assume A is empty. This means that there are no elements in A.
Now, let's consider the cross product A cross (C cross B). By definition, the cross product of two sets A and B is the set of all possible ordered pairs (a, b) where a is an element of A and b is an element of B. Since A is empty, there are no elements in A to form any ordered pairs. Therefore, A cross (C cross B) will also be empty.
Next, we need to prove that C cross B is empty if and only if B is empty.
Assume C cross B is empty. This means that there are no elements in C cross B, and hence, no ordered pairs can be formed. If C cross B is empty, it implies that C is also empty because if C had any elements, we could form ordered pairs with those elements and elements from B.
Now, if C is empty, then it follows that B must also be empty. If B had any elements, we could form ordered pairs with those elements and elements from the empty set C, contradicting the assumption that C cross B is empty.
Therefore, we have shown that if A is empty, then A cross (C cross B) if and only if C cross B is empty, which can also be written as CC B.
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Select the correct answer.
Which of the following represents a factor from the expression given?
5(3x² +9x) -14
O 15x²
O5
O45x
O 70
The factor from the expression 5(3x² + 9x) - 14 is not listed among the options you provided. However, I can help you simplify the expression and identify the factors within it.
To simplify the expression, we can distribute the 5 to both terms inside the parentheses:
5(3x² + 9x) - 14 = 15x² + 45x - 14
From this simplified expression, we can identify the factors as follows:
15x²: This represents the term with the variable x squared.
45x: This represents the term with the variable x.
-14: This represents the constant term.
Therefore, the factors from the expression are 15x², 45x, and -14.
Product, Quotient, Chain rules and higher Question 2, 1.6.3 Part 1 of 3 a. Use the Product Rule to find the derivative of the given function. b. Find the derivative by expanding the product first. f(x)=(x-4)(4x+4) a. Use the product rule to find the derivative of the function. Select the correct answer below and fill in the answer box(es) to complete your choice. OA. The derivative is (x-4)(4x+4) OB. The derivative is (x-4) (+(4x+4)= OC. The derivative is x(4x+4) OD. The derivative is (x-4X4x+4)+(). E. The derivative is ((x-4). HW Score: 83.52%, 149.5 of Points: 4 of 10
The derivative of the function f(x) = (x - 4)(4x + 4) can be found using the Product Rule. The correct option is OC i.e., the derivative is 8x - 12.
To find the derivative of a product of two functions, we can use the Product Rule, which states that the derivative of the product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).
Applying the Product Rule to the given function f(x) = (x - 4)(4x + 4), we differentiate the first function (x - 4) and keep the second function (4x + 4) unchanged, then add the product of the first function and the derivative of the second function.
a. Using the Product Rule, the derivative of f(x) is:
f'(x) = (x - 4)(4) + (1)(4x + 4)
Simplifying this expression, we have:
f'(x) = 4x - 16 + 4x + 4
Combining like terms, we get:
f'(x) = 8x - 12
Therefore, the correct answer is OC. The derivative is 8x - 12.
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In Problems 1 through 12, verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with re- spect to x. 1. y' = 3x2; y = x³ +7 2. y' + 2y = 0; y = 3e-2x 3. y" + 4y = 0; y₁ = cos 2x, y2 = sin 2x 4. y" = 9y; y₁ = e³x, y₂ = e-3x 5. y' = y + 2e-x; y = ex-e-x 6. y" +4y^ + 4y = 0; y1= e~2x, y2 =xe-2x 7. y" - 2y + 2y = 0; y₁ = e cos x, y2 = e* sinx 8. y"+y = 3 cos 2x, y₁ = cos x-cos 2x, y2 = sinx-cos2x 1 9. y' + 2xy2 = 0; y = 1+x² 10. x2y" + xy - y = ln x; y₁ = x - ln x, y2 = =-1 - In x In x 11. x²y" + 5xy' + 4y = 0; y1 = 2 2 = x² 12. x2y" - xy + 2y = 0; y₁ = x cos(lnx), y2 = x sin(In.x)
The solutions to the given differential equations are:
y = x³ + 7y = 3e^(-2x)y₁ = cos(2x), y₂ = sin(2x)y₁ = e^(3x), y₂ = e^(-3x)y = e^x - e^(-x)y₁ = e^(-2x), y₂ = xe^(-2x)y₁ = e^x cos(x), y₂ = e^x sin(x)y₁ = cos(x) - cos(2x), y₂ = sin(x) - cos(2x)y = 1 + x²y₁ = x - ln(x), y₂ = -1 - ln(x)y₁ = x², y₂ = x^(-2)y₁ = xcos(ln(x)), y₂ = xsin(ln(x))To verify that each given function is a solution of the given differential equation, we will substitute the function into the differential equation and check if it satisfies the equation.
1. y' = 3x²; y = x³ + 7
Substituting y into the equation:
y' = 3(x³ + 7) = 3x³ + 21
The derivative of y is indeed equal to 3x², so y = x³ + 7 is a solution.
2. y' + 2y = 0; y = 3e^(-2x)
Substituting y into the equation:
y' + 2y = -6e^(-2x) + 2(3e^(-2x)) = -6e^(-2x) + 6e^(-2x) = 0
The equation is satisfied, so y = 3e^(-2x) is a solution.
3. y" + 4y = 0; y₁ = cos(2x), y₂ = sin(2x)
Taking the second derivative of y₁ and substituting into the equation:
y"₁ + 4y₁ = -4cos(2x) + 4cos(2x) = 0
The equation is satisfied for y₁.
Taking the second derivative of y₂ and substituting into the equation:
y"₂ + 4y₂ = -4sin(2x) - 4sin(2x) = -8sin(2x)
The equation is not satisfied for y₂, so y₂ = sin(2x) is not a solution.
4. y" = 9y; y₁ = e^(3x), y₂ = e^(-3x)
Taking the second derivative of y₁ and substituting into the equation:
y"₁ = 9e^(3x)
9e^(3x) = 9e^(3x)
The equation is satisfied for y₁.
Taking the second derivative of y₂ and substituting into the equation:
y"₂ = 9e^(-3x)
9e^(-3x) = 9e^(-3x)
The equation is satisfied for y₂.
5. y' = y + 2e^(-x); y = e^x - e^(-x)
Substituting y into the equation:
y' = e^x - e^(-x) + 2e^(-x) = e^x + e^(-x)
The equation is satisfied, so y = e^x - e^(-x) is a solution.
6. y" + 4y^2 + 4y = 0; y₁ = e^(-2x), y₂ = xe^(-2x)
Taking the second derivative of y₁ and substituting into the equation:
y"₁ + 4(y₁)^2 + 4y₁ = 4e^(-4x) + 4e^(-4x) + 4e^(-2x) = 8e^(-2x) + 4e^(-2x) = 12e^(-2x)
The equation is not satisfied for y₁, so y₁ = e^(-2x) is not a solution.
Taking the second derivative of y₂ and substituting into the equation:
y"₂ + 4(y₂)^2 + 4y₂ = 2e^(-2x) + 4(xe^(-2x))^2 + 4xe^(-2x) = 2e^(-2x) + 4x^2e^(-4x) + 4xe^(-2x)
The equation is not satisfied for y₂, so y₂ = xe^(-2x) is not a solution.
7. y" - 2y + 2y = 0; y₁ = e^x cos(x), y₂ = e^x sin(x)
Taking the second derivative of y₁ and substituting into the equation:
y"₁ - 2(y₁) + 2y₁ = e^x(-cos(x) - 2cos(x) + 2cos(x)) = 0
The equation is satisfied for y₁.
Taking the second derivative of y₂ and substituting into the equation:
y"₂ - 2(y₂) + 2y₂ = e^x(-sin(x) - 2sin(x) + 2sin(x)) = 0
The equation is satisfied for y₂.
8. y" + y = 3cos(2x); y₁ = cos(x) - cos(2x), y₂ = sin(x) - cos(2x)
Taking the second derivative of y₁ and substituting into the equation:
y"₁ + y₁ = -cos(x) + 2cos(2x) + cos(x) - cos(2x) = cos(x)
The equation is not satisfied for y₁, so y₁ = cos(x) - cos(2x) is not a solution.
Taking the second derivative of y₂ and substituting into the equation:
y"₂ + y₂ = -sin(x) + 2sin(2x) + sin(x) - cos(2x) = sin(x) + 2sin(2x) - cos(2x)
The equation is not satisfied for y₂, so y₂ = sin(x) - cos(2x) is not a solution.
9. y' + 2xy² = 0; y = 1 + x²
Substituting y into the equation:
y' + 2x(1 + x²) = 2x³ + 2x = 2x(x² + 1)
The equation is satisfied, so y = 1 + x² is a solution.
10 x²y" + xy' - y = ln(x); y₁ = x - ln(x), y₂ = -1 - ln(x)
Taking the second derivative of y₁ and substituting into the equation:
x²y"₁ + xy'₁ - y₁ = x²(0) + x(1) - (x - ln(x)) = x
The equation is satisfied for y₁.
Taking the second derivative of y₂ and substituting into the equation:
x²y"₂ + xy'₂ - y₂ = x²(0) + x(-1/x) - (-1 - ln(x)) = 1 + ln(x)
The equation is not satisfied for y₂, so y₂ = -1 - ln(x) is not a solution.
11. x²y" + 5xy' + 4y = 0; y₁ = x², y₂ = x^(-2)
Taking the second derivative of y₁ and substituting into the equation:
x²y"₁ + 5xy'₁ + 4y₁ = x²(0) + 5x(2x) + 4x² = 14x³
The equation is not satisfied for y₁, so y₁ = x² is not a solution.
Taking the second derivative of y₂ and substituting into the equation:
x²y"₂ + 5xy'₂ + 4y₂ = x²(4/x²) + 5x(-2/x³) + 4(x^(-2)) = 4 + (-10/x) + 4(x^(-2))
The equation is not satisfied for y₂, so y₂ = x^(-2) is not a solution.
12. x²y" - xy' + 2y = 0; y₁ = xcos(ln(x)), y₂ = xsin(ln(x))
Taking the second derivative of y₁ and substituting into the equation:
x²y"₁ - xy'₁ + 2y₁ = x²(0) - x(-sin(ln(x))/x) + 2xcos(ln(x)) = x(sin(ln(x)) + 2cos(ln(x)))
The equation is satisfied for y₁.
Taking the second derivative of y₂ and substituting into the equation:
x²y"₂ - xy'₂ + 2y₂ = x²(0) - x(cos(ln(x))/x) + 2xsin(ln(x)) = x(sin(ln(x)) + 2cos(ln(x)))
The equation is satisfied for y₂.
Therefore, the solutions to the given differential equations are:
y = x³ + 7
y = 3e^(-2x)
y₁ = cos(2x)
y₁ = e^(3x), y₂ = e^(-3x)
y = e^x - e^(-x)
y₁ = e^(-2x)
y₁ = e^x cos(x), y₂ = e^x sin(x)
y = 1 + x²
y₁ = xcos(ln(x)), y₂ = xsin(ln(x))
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