Beginner Differential Equations

Hello,

I've just started on separating differential equations and am just having a little trouble on how to proceed with this sort of question.

$\displaystyle \frac {dy}{dx} = y^2 \,e^(2x)$

I understand that the y needs to be bough to the left, and then the formal separation of $\displaystyle \frac {dy}{dx}$ occurs - as my tutor puts it.

$\displaystyle =\frac{1}{y^2}\,dy = e^{2x} dx$

$\displaystyle = \int \frac{1}{y^2}dy = \int e^{2x}$

$\displaystyle = ln\,y^2 = \frac {e^{2x}}{2} + C$

This is where it breaks down a little bit for me...I'd assume that we need to get y to its simplest form, but I don't know how to do that...

$\displaystyle =y^2 = e^{\frac {e^2x}{2}} + C$

That isn't right (I'm assuming), so can someone point me in the direction of the correct step.

Thanks :)

Re: Beginner Differential Equations

Your seperation is good but

$\displaystyle \int \frac{1}{y^2}dy = -\frac{1}{y}$

So you should get

$\displaystyle -\frac{1}{y}=\frac{1}{2}e^{2x}+C$

Re: Beginner Differential Equations

First, when you separate the variables to get:

$\displaystyle \frac{1}{y^2}\cdot\frac{dy}{dx}=e^{2x}$

you should note that you are losing the trivial solution $\displaystyle y\equiv0$.

Then integrating with respect to $\displaystyle x$, we have:

$\displaystyle \int y^{-2}\,dy=\int e^{2x}\,dx$

As noted above, you have integrated incorrectly.

Re: Beginner Differential Equations

Quote:

Originally Posted by

**TheEmptySet** Your seperation is good but

$\displaystyle \int \frac{1}{y^2}dy = -\frac{1}{y}$

So you should get

$\displaystyle -\frac{1}{y}=\frac{1}{2}e^{2x}+C$

Ahh, of course...It's basically $\displaystyle y^{-2}$.

I was treating it as $\displaystyle y^{-1}$ or something like that.

Thank you!! :)