# Finding a solution given initial value problem

• Sep 22nd 2012, 07:55 AM
Finding a solution given initial value problem
Code:

y'-y=7te2t  , y(0)=1

I would like to know how to solve this Differential Equation. I think I have to find the integrating factor.

It does look like it follows the the standard form
Code:

y'+px=qx
So the integrating factor is e∫px

Would the integrating factor be e∫-y ?

Thanks
• Sep 22nd 2012, 08:16 AM
MarkFL
Re: Finding a solution given initial value problem

$\mu(t)=e^{\int(-1)\,dt}=e^{-t}$

Multiplying the ODE by this factor will give you the product of differentiation on the left:

$e^{-t}\frac{dy}{dt}-e^{-t}y=7te^t$

$\frac{d}{dt}\left(e^{-t}y \right)=7te^t$

$\int\,d\left(e^{-t}y \right)=7\int te^t\,dt$

Now, use integration by parts on the right...
• Sep 22nd 2012, 12:41 PM
$\int_a^b f(x)\,dx=F(b)-F(a)$