Question
Asked Sep 6, 2019
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Find the general solution of the differential equation

y' + y = te-t + et

and use it to determine how solutions behave as t → ∞.

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Expert Answer

Step 1

The differential equation is given as

yytee
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yytee

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Step 2

Use method of integrating factor such that the given differential equation is compared to the general equation:

y'P(t)y(t)
[(t)e!4
The solution is given as ye2
dt C
Here, P(t1 and Q(t) =te" +e'
The integrating factor is obtained as e*i e.
ldt
= e
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y'P(t)y(t) [(t)e!4 The solution is given as ye2 dt C Here, P(t1 and Q(t) =te" +e' The integrating factor is obtained as e*i e. ldt = e

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Step 3

Obtain the solution...

C
ye (t+e)dt +C
t2e2
C
2
ye'
2
2r
C
(Divide by e'
y =
2e 2e e'
t2e*
That is, y2e
+
e'
2
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C ye (t+e)dt +C t2e2 C 2 ye' 2 2r C (Divide by e' y = 2e 2e e' t2e* That is, y2e + e' 2

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Advanced Math