Finding the solution to an IVP Problem. Basic Differential Equations problem.

[cyturk]

I am trying to solve an IVP problem and I seem to be stuck on it because I am getting an integration that seems very complicated and I think I messed up on it, I have my work so far below.

Homework Statement

Find the solution to the IVP

ty^'+7y=2t^2 e^2t, y(1)=7

Is this equation linear? Determine in what interval the solutions exist.

Homework Equations

The Attempt at a Solution

The image has my work so far, as you can see the integration for (t^7)(2te^2t) is a beast and that is why I think I am wrong so far. Here is the integration answer http://www.wolframalpha.com/input/?i=integrate+%28t^7%29%282te^%282t%29%29.

 

[JJacquelin]

The primitives of (t^8)exp(2t) are on the form P(x)exp(2t) were P(x) is a 8th degree polynomial.
Derive this function and find the coefficients of the polynomial by indentification with (t^8)exp(2t)

 

[hunt_mat]

You started out very well, and from:
$$\frac{d}{dt}(t^{7}y)=2t^{8}e^{2t}$$
I think you have made an error, you can do two things: 1) You can do an indefinite integration and add an integration constant and find that constand by using the initial condition or 2) integrate from 1 to t both sides and use Y(1)=7.

 

[HallsofIvy]

Integrate 2t⁸e^2t using integration by parts- 8 times!

 

[cyturk]

Integrate 2t⁸e^2t using integration by parts- 8 times!

Thanks for everyone else and this is what I ended up doing. It was the correct way of solving the problem even though it was a little bit of a hassle! :)