Question

Solve the differential equation

Answer 1

\[y'(t)-2e^t \int\limits_{0}^{t}e^{-\xi}y(\xi )d \xi =t\]

Answer 2

y(0)=2

Answer 3

I know this requires convolution integrals but that integral term is bothering me.

Answer 4

It's a function times an integral. How would I even deal with that? :/ .

Answer 5

I want to say the e^(t) function does not include the variable which is used in the integration so it can be taken into and out of the integrand freely.

Answer 6

But then we have three terms inside that integral :/ .

Answer 7

Well
\( e^{-\xi} e^{t} y(\xi) d \xi \) = \( e^{t - \xi} y(\xi) d \xi \) by the property of multiplying like-bases.

Answer 8

yep I see that.

Answer 9

that is then the definition of a convolution, correct?

Answer 10

Ooo!!! How did I not see that? >.< . Thanks!

Answer 11

You're welcome! :D