Veryfing ODE for complicated y(t)

rdioface

1. The problem statement, all variables and given/known data
For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution.


2. Relevant equations
[itex] y' - 4ty = 1 [/itex]

[itex] y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds[/itex]

3. The attempt at a solution
I attempted to take [itex]\frac{d}{dt}[/itex] of y(t) as usual but
1. if I do not try bringing the [itex]\frac{d}{dt}[/itex] inside the integral I can do nothing because there is no elementary antiderivative of y(t).
2. if I do bring the [itex]\frac{d}{dt}[/itex] inside the integral, I can use the chain rule to get
[itex] y(t) = (-2) \int_{0}^{t} (-2t) e^{-2(s^{2}-t^{2})} ds[/itex]
but since my variable of integration is ds not dt, this doesn't allow me to use a u-substitution as I had hoped nor can I think of a way to relate ds and dt.

More or less I do not know how to take [itex]\frac{d}{dt}[/itex] of y(t) and I do not know any other ways to solve the problem.

τheory

Do you have to do it that way? Because it looks like you can solve it linearly.

rdioface

Yes, the problem asks to verify that y(t) is a solution to the differential equation y' + 4ty = 1.

Dick

I don't think you have to be able to do the integral to identify that expression as [itex]4 t y(t)[/itex]. Since you are integrating ds you can factor out the t.

rdioface

facepalm.jpg

Thanks!