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

1. Jun 12, 2011

### 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.

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

2. Jun 13, 2011

### 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)

3. Jun 13, 2011

### 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.

4. Jun 13, 2011

### HallsofIvy

Staff Emeritus
Integrate 2t8e2t using integration by parts- 8 times!

5. Jun 26, 2011

### cyturk

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! :)