Finding a solution given initial value problem

Code:

`y'-y=7te`^{2t ,} 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 So the integrating factor is e^{∫px }Would the integrating factor be e^{∫-y} ?

Thanks

Re: Finding a solution given initial value problem

Your integrating factor would be:

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

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

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

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

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

Now, use integration by parts on the right...

Re: Finding a solution given initial value problem

How do you insert mathematical functions on these forums the way you inserted the integral sign?

And thanks for the integrating factor.

Re: Finding a solution given initial value problem

Use the TEX tags, for example enclosing the code \int_a^b f(x)\,dx=F(b)-F(a) within the tags produces:

$\displaystyle \int_a^b f(x)\,dx=F(b)-F(a)$