Recall that variation of parameters is used to solve second-order ODEs of the form
y''(t) + p(t) y'(t) + q(t) y(t) = f(t)
so the first thing you need to do is divide both sides of your equation by t :
y'' + (2t - 1)/t y' - 2/t y = 7t
You're looking for a solution of the form
[tex]y=y_1u_1+y_2u_2[/tex]
where
[tex]u_1(t)=\displaystyle-\int\frac{y_2(t)f(t)}{W(y_1,y_2)}\,\mathrm dt[/tex]
[tex]u_2(t)=\displaystyle\int\frac{y_1(t)f(t)}{W(y_1,y_2)}\,\mathrm dt[/tex]
and W denotes the Wronskian determinant.
Compute the Wronskian:
[tex]W(y_1,y_2) = W\left(2t-1,e^{-2t}\right) = \begin{vmatrix}2t-1&e^{-2t}\\2&-2e^{-2t}\end{vmatrix} = -4te^{-2t}[/tex]
Then
[tex]u_1=\displaystyle-\int\frac{7te^{-2t}}{-4te^{-2t}}\,\mathrm dt=\frac74\int\mathrm dt = \frac74t[/tex]
[tex]u_2=\displaystyle\int\frac{7t(2t-1)}{-4te^{-2t}}\,\mathrm dt=-\frac74\int(2t-1)e^{2t}\,\mathrm dt=-\frac74(t-1)e^{2t}[/tex]
The general solution to the ODE is
[tex]y = C_1(2t-1) + C_2e^{-2t} + \dfrac74t(2t-1) - \dfrac74(t-1)e^{2t}e^{-2t}[/tex]
which simplifies somewhat to
[tex]\boxed{y = C_1(2t-1) + C_2e^{-2t} + \dfrac74(2t^2-2t+1)}[/tex]
Helppp and explain!!!
A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now. What will be the amount of the material right after 20 years
Answer:
x = 960.4
Step-by-step explanation:
980 = 1000[tex]e^{kt}[/tex]
.98 = [tex]e^{10 k}[/tex]
ln(.98) = 10k ln(e)
k = ln(.98)/10
k=-0.00202
~~~~~~~~~~~~~~
x = 1000[tex]e^{20 *-.00202}[/tex]
x = 960.4
The amount of the material right after 20 years will be x = 960.4.
What is an exponential expression?Powers can simply be expressed in concise form using exponential expressions. The exponent shows how many times the base has been multiplied. Since 2 is the "base" and 5 is the "exponent," it can be represented as 2x2x2x2=25 for the number 32. This phrase should be understood as "two to the fifth power."
Given that radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 1000 grams of the material 10 years ago. There are 980 grams right now.
The amount of the material will be calculated as,
980 = 1000
[tex]0.98 = e^{10k}[/tex]
ln(.98) = 10k ln(e)
k = ln(.98)/10
k=-0.00202
The value after 20 years will be,
[tex]x = 1000e^{20\times 0.00202}[/tex]
x = 960.4
Therefore, the amount of the material right after 20 years will be x = 960.4.
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Two cities,a and are mapped on the coordinate plane. How far apart are they from each other?
Answer:
[tex]\sqrt{97} \\ \sqrt{9^{2}+4^{2} }[/tex]
Step-by-step explanation:
Hw help ASAP PLZZZZZZ
Answer:
Your answer is C. X = 29/8c
Step-by-step explanation:
2/3(cx + 1/2) - 1/4 = 5/2
2cx/3+1/3-1/4=5/2
2cx3+1/12=5/2
2cx/3=5/2-1/12
2cx/3=29/12
(3)2cx/3=29/12(3)
2cx= 31/4
(2c)2cx=29/4(2c)
X=29/8c
Your answer is C. X = 29/8c
A field book is a private notepad used by a surveyor to transcribe notes and is not considered a legal document True False
Answer:
False
Step-by-step explanation:
A field book is a private notepad used by a surveyor to record measurements and notes.
Basically, the size of a field book is 200 millimeters × 120 millimeters (20 centimeters × 12 centimeters) and it's typically opened lengthwise. There are two (2) main types of field book and these includes;
I. Double-line field book.
II. Single-line field book.
As a general rule, it's best that all findings, entries (notes) and observations are recorded or made into a field book after each and every measurement have been taken by a surveyor.
In conclusion, a field book is considered to be a legal document used by surveyors to keep records of accomplished field work or work done in the field. Thus, it's not a private notepad used by a surveyor to transcribe notes.
Answer:
False
Step-by-step explanation:
A field book is a private notepad used by a surveyor to transcribe notes and is not considered a legal document is False.
Test scores for a Statistics class have a mean of 78 with a standard deviation of 6. Suppose a student gets an 81 on that test. What is the z-score for that grade?
Answer:
0.3209
in case you dont know A z-score tells you if the distribution it comes from is normal.
Step-by-step explanation:
3.
If 9x = 27, what is the value of x?
Tell whether the following two triangles can be
proven congruent through SAS.
A.Yes, the two triangles are congruent
because two sides and their included
angle are congruent in both triangles.
B.No, the two triangles don't have
corresponding sides marked congruent.
C. Yes, the two triangles are congruent because they’re both right triangles.
D.No, the two triangles can only be proven congruent through SSA.
Answer:
B. No, the two triangles don't have
corresponding sides marked congruent.
oludonts c) 2x + y = 2
2x + 2y = 0
Answer:
[tex]x = 2[/tex]
[tex]y = -2[/tex]
Step-by-step explanation:
Given
[tex]2x + y = 2[/tex]
[tex]2x + 2y = 0[/tex]
Required
Solfe for x and y
Subtract both equations
[tex]2x- 2x + y - 2y = 2 -0[/tex]
[tex]-y = 2[/tex]
Divide by -1
[tex]y = -2[/tex]
Substitute [tex]y = -2[/tex] in [tex]2x + 2y = 0[/tex]
[tex]2x+2 *-2 = 0[/tex]
[tex]2x-4 = 0[/tex]
[tex]x - 2 = 0[/tex]
Collect like terms
[tex]x = 2[/tex]
1). A population of 20 rabbits is released into a wildlife region. The population is growing at a rate of 60% per year.
A) the General Equation from the Video was: P(x) = (blank)
What is the population of rabbits after 5 years?
B) the Evaluated equation I used to get the following answer is (blank)
and After five years there will be(blank)
rabbits.
And What is the population of rabbits after 8 years?
c) the Evaluated equation I used to get the following answer is(blank)
and After eight years there will be(blank)
rabbits.
Answer:
(a) A = 20(1.6)^t
(b) 210 rabbits
Step-by-step explanation:
Initial number of rabbits = 20
rate of growth, R = 60 % annually
(A) The general equation is
[tex]A = P \left ( 1+\frac{R}{100} \right )^t\\\\A = 20\left ( 1+\frac{60}{100} \right )^t\\\\A = 20 (1.6)^t[/tex]
(B) Let the time, t = 5 years
So, the population after 5 years is
[tex]A = 20 (1.6)^5\\\\A = 209.7 = 210 rabbits[/tex]
Jen recently rode her bicycle to visit her friend who lives 6 miles away. On her way there, her average speed was & miles per hour faster than on her way home. If jen
spent a total of 2 hours bicycling find the two rates.
Answer:
(1) +t+=+1.2+ hrs
Step-by-step explanation:
To determine the organic material in a dried lake bed, the percent carbon by mass is measured at two different locations. To compare the means of the two different locations, it must first be determined whether the standard deviations of the two locations are different. For each location, calculate the standard deviation and report it with two significant figures.
Answer:
[tex]\sigma_1 = 0.08[/tex] --- Location 1
[tex]\sigma_2 = 0.34[/tex] --- Location 2
Step-by-step explanation:
Given
See attachment for the given data
Required
The standard deviation of each location
For location 1
First, calculate the mean
[tex]\bar x_1 =\frac{\sum x}{n}[/tex]
So, we have:
[tex]\bar x_1 =\frac{30.40+30.20+30.30+30.40+30.30}{5}[/tex]
[tex]\bar x_1 =\frac{151.60}{5}[/tex]
[tex]\bar x_1 =30.32[/tex]
The standard deviation is calculated as:
[tex]\sigma_1 = \sqrt{\frac{\sum(x - \bar x_1)^2}{n-1}}[/tex]
[tex]\sigma_1 = \sqrt{\frac{(30.40 - 30.32)^2+(30.20 - 30.32)^2+(30.30 - 30.32)^2+(30.40 - 30.32)^2+(30.30 - 30.32)^2}{5-1}}[/tex]
[tex]\sigma_1 = \sqrt{\frac{0.028}{4}}[/tex]
[tex]\sigma_1 = \sqrt{0.007}[/tex]
[tex]\sigma_1 = 0.08[/tex]
For location 2
First, calculate the mean
[tex]\bar x_2 =\frac{\sum x}{n}[/tex]
So, we have:
[tex]\bar x_2 =\frac{30.10+30.90+30.20+30.70+30.30}{5}[/tex]
[tex]\bar x_2 =\frac{152.2}{5}[/tex]
[tex]\bar x_2 =30.44[/tex]
The standard deviation is calculated as:
[tex]\sigma_2 = \sqrt{\frac{\sum(x - \bar x_2)^2}{n-1}}[/tex]
[tex]\sigma_2 = \sqrt{\frac{(30.10-30.44)^2+(30.90-30.44)^2+(30.20-30.44)^2+(30.70-30.44)^2+(30.30-30.44)^2}{5-1}}[/tex]
[tex]\sigma_2 = \sqrt{\frac{0.472}{4}}[/tex]
[tex]\sigma_2 = \sqrt{0.118}[/tex]
[tex]\sigma_2 = 0.34[/tex]
Which one is a better deal?
a 3 pack of notebooks for $4.50
Or a 12 pack of notebooks for $15.00
determine lcm and HCF of 24 and 26 using prime factors
Answer:
WHAT IS THE FACTOR?
Step-by-step explanation:
What is the median of the following set of values? 7, 21, 19, 15, 19, 14, 15, 19
The median of the following set of values is equals to 17.
What are median?Median represents the middle value of the given data when arranged in a particular order. The mean is the average value which can be calculated by dividing the sum of observations by the number of observations
We are given that the median of the following set of values
7, 21, 19, 15, 19, 14, 15, 19
Line them up in order first.
7, 14, 15, 15, 19, 19, 19, 21
Here the middle value are 15 and 19.
The median is 15 and 19. OR 17,
Therefore, 15 + 19 = 34/2 which equals to 17.
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.It is 12:00 and people are lining up for the matinee at the Bijou Cinema Six. In the first five minutes (12:05), 6 people get into line. At the end of the second five minutes (12:10), there are 11 people in line. At the end of the third five minutes (12:15) there are 16 people in line. If the people keep lining up at this rate, what time will it be when there are 81 people in line?
Answer:
65 minutes
Step-by-step explanation:
already 16 people in line
total is 81
81 - 16 equals 65
about 5 people every 5 minutes (basically 1 min per person)
so, answer probably 65
Solve each inequality. Graph the solution on a number line.
Answer:
n>2 2/3
Draw a filled dot at a little more than 2 1/2 and continue the line to the right.
Step-by-step explanation:
Subtract 1/3 from both sides to get
2 2/3
Flip the inequality
n> 2 2/3
I hope this helps!
PLEASE I NEED HELP!!
Answer:
it is (4,120)
hope this helps you
What is the value of the expression [-7] + [-4]
Answer:
11
Step-by-step explanation:
I'm assuming that [.] fldenote absolute value even tho the absolute value function is represented by (|.|)
value of [-7] will be positive that us 7.
= 7 + 4
= 11
Find the area of the figure
Please help :)
9514 1404 393
Answer:
66.5 cm²
Step-by-step explanation:
A horizontal line at the "knee" on the right will divide the figure into a 4 cm by 2 cm rectangle, and a trapezoid with bases 4 cm and 9 cm, and height 11-2 = 9 cm. Then the total area of the figure is ...
A = LW + 1/2(b1 +b2)h
A = (4 cm)(2 cm) + (1/2)(4 cm +9 cm)(9 cm) = 8 cm² +58.5 cm²
A = 66.5 cm² . . . . area of the figure
A manufacturer inspects a sample of 500 smart phones and finds that 496 of them have no defects. The manufacturer sent a shipment of 2000 smartphones to a distributor. Predict the number of smartphones in the shipment that are likely to have no dects.
Answer:
1984
Step-by-step explanation:
What are the coordinates of vertex F" of ΔF"G"H"?
) A patient drank 12 ounces of orange juice. How many milliliters did the patient drink?
Answer:
[tex]Drink = 354.882\ mL[/tex]
Step-by-step explanation:
Given
[tex]Drink = 12oz[/tex]
Required
Equivalent in mL
We have:
[tex]1\ oz = 29.5735\ mL[/tex]
So:
[tex]Drink = 12 * 29.5735mL[/tex]
[tex]Drink = 354.882\ mL[/tex]
Help pls ty!
Adios!
Bye
If there is a 90% chance of rain tomorrow and a 70% chance of wind and rain,
what is the probability that it is windy, given that it is rainy? Round your
answer to the nearest percent.
Answer:
78%
Step-by-step explanation:
the chance of rain
P(R) = 0.9
the chance of wind and rain
P(R∩ W) = 0.7
The probabilities of windy weather is
P(W) = [tex]\frac{P(RW)}{P(R)} =\frac{0.7}{0.9} =0.78[/tex]
The correct answer is B) 78%
Please give me a 100% correct answer
A ship sailed 30 kilometers in 172 hours. What was its rate in kilometers per hour?
(1) 20
(2) 30
(3) 45
(4) 90
(5) Not enough information is given.
Answer:
See edit
A 20 km / hour
Step-by-step explanation:
The exact answer is 30 km / 172 which is less than 1 km / hr. Since that answer isn't offered, I suspect there is something wrong with the question. If there is a decimal after the 1 in 172 you would get 17.44 km / hr. That's roughly A.
If the decimal is not there, I think you should either resubmit the question or answer A.
Edit
The note said (below) that the ship went 30 km in 1 1/2 hours.
Rate = distance / time
distance = 30 km
time = 1 1/2 hours = 1.5 hours
rate = 30 / 1.5 = 20 km / hr
y is inversely proportional to x when y=9, x=24
Answer:
216
Step-by-step explanation:
y=k÷x
k=xy
k=9×24
k=216
suppose △abc≅△xyz. what is the corresponding congruent part for each segment or angle?
Answer:
Step-by-step explanation:
37. Two numbers are such that their
difference, their sum and their
product are in the ratio 1:7: 24.
Find the product of the
number.
Answer:
8 and 6
Step-by-step explanation:
Two numbers are such that their difference, their sum, and their product are to
each other as 1:7:24. Their product must equal what number?
:
Two numbers a & b
Let x = the multiplier
:
a - b = 1x
a + b = 7x
a * b = 24x
:
Add the 1st two equations
a - b = x
a + b = 7x
2a = 8x
a = 4x
or
x = .25a
:
a * b = 24x
Replace 24x; a = 4x therefore:
a * b = 6a
b = 6
;
Using the 1st equation
a - b = 1x
Replace b with 6 and x with .25a
a - 6 = .25a
a - .25a = 6
.75a = 6
a =
a = 8
:
Find the multiplier
a - b = x
8 - 6 = 2
:
Check this
a - b = 2 (1*2)
a + b = 14; (7*2)
a * b = 48: (24*2)
:
The numbers are 8 and 6; their products = 48
Kala drove 819 miles in 13 hours.
At the same rate, how long would it take her to drive 441 miles?
Answer:
[tex]{ \tt{ \frac{819}{13} = \frac{441}{h} }} \\ { \tt{h = \frac{(441 \times 13)}{819} }} \\ h = 7 \: hours[/tex]
