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Mars2501 [29]
3 years ago
9

Consider the initial value problem y′+2y=4t,y(0)=8.

Mathematics
1 answer:
Xelga [282]3 years ago
5 0

Answer:

Please read the complete procedure below:

Step-by-step explanation:

You have the following initial value problem:

y'+2y=4t\\\\y(0)=8

a) The algebraic equation obtain by using the Laplace transform is:

L[y']+2L[y]=4L[t]\\\\L[y']=sY(s)-y(0)\ \ \ \ (1)\\\\L[t]=\frac{1}{s^2}\ \ \ \ \ (2)\\\\

next, you replace (1) and (2):

sY(s)-y(0)+2Y(s)=\frac{4}{s^2}\\\\sY(s)+2Y(s)-8=\frac{4}{s^2}  (this is the algebraic equation)

b)

sY(s)+2Y(s)-8=\frac{4}{s^2}\\\\Y(s)[s+2]=\frac{4}{s^2}+8\\\\Y(s)=\frac{4+8s^2}{s^2(s+2)} (this is the solution for Y(s))

c)

y(t)=L^{-1}Y(s)=L^{-1}[\frac{4}{s^2(s+2)}+\frac{8}{s+2}]\\\\=L^{-1}[\frac{4}{s^2(s+2)}]+L^{-1}[\frac{8}{s+2}]\\\\=L^{-1}[\frac{4}{s^2(s+2)}]+8e^{-2t}

To find the inverse Laplace transform of the first term you use partial fractions:

\frac{4}{s^2(s+2)}=\frac{-s+2}{s^2}+\frac{1}{s+2}\\\\=(\frac{-1}{s}+\frac{2}{s^2})+\frac{1}{s+2}

Thus, you have:

y(t)=L^{-1}[\frac{4}{s^2(s+2)}]+8e^{-2t}\\\\y(t)=L^{-1}[\frac{-1}{s}+\frac{2}{s^2}]+L^{-1}[\frac{1}{s+2}]+8e^{-2t}\\\\y(t)=-1+2t+e^{-2t}+8e^{-2t}=-1+2t+9e^{-2t}  

(this is the solution to the differential equation)

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PLEASE HELP BEEN STUCK FOR HOURS WILL MARK THE BRAINLIEST<br> HAVE TO ANSWER A,B, and C
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Answer:

A. 1)5.5 2)4.5 3)5.75 4)6.25 B. 1.75 C. A correct B incorrect C incorrect D correct

Step-by-step explanation:

Mean is a fancy way of saying <em><u>Average </u></em>, when calculating average you add all the numbers and divide by how many numbers there are:

<em><u>A</u></em>

<u>1.</u>

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<u>2.</u>

5+7+4+2=18/4=4.5

<u>3.</u>

8+7+5+3=23/4=5.75

<u>4.</u>

6+8+4+5=25/4=6.25

<em><u>B</u></em>

largest mean is 6.25, smallest mean is 4.5:

<u>Subtract lowest mean from highest mean:</u>

6.25-4.5=1.75

<em><u>C</u></em>

<u>A)</u>

this is correct because averaging your average gives an overall average,

<u>B)</u>

if a is correct b has to be incorrect

<u>C)</u>

if D is correct C must be incorrect

<u>D)</u>

The closer they are together, that means more likely to be accurate because they are almost the same

I hope this helps u!

pls pls PLS give a brainliest and a thx ;)

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