Finding a solution given initial value problem

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September 22nd 2012, 08:55 AM
terrygrada

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

Code:

y'+px=qx

So the integrating factor is e^∫pxWould the integrating factor be e^∫-y ?

Thanks
September 22nd 2012, 09:16 AM
MarkFL

Re: Finding a solution given initial value problem

Your integrating factor would be:

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

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...
September 22nd 2012, 01:41 PM
terrygrada

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.
September 22nd 2012, 02:04 PM
MarkFL

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:

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

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