# Thread: Can someone help with and explain this differential equation?

1. ## Can someone help with and explain this differential equation?

I know it looks easy, but my working does not look right!

(x² + 1) dy/dx - 2xy = 2x (x² + 1)

1) which analytical method would you use to solve this equation?
and therefore solve expressing y explicitly as a function of x.
Hence find the particular solution of the differential equation that satisfies the initial condition y(0) = 1

Many thanks

2. Hi. I would divide through by $\displaystyle (x^2+1)$. That puts it into standard form of a first order ode. Now calculate the integrating factor and proceed.

3. I am working on this similar problem - I think I have got to the last step and am trying to integrate the following...

$\displaystyle \int e^{x^2}2x dx$

Should be integrating this by parts?

I have seen in another text where this has been integrated it in one go but I don't know if this is right?

4. You can integrate this by substitution, let:

$\displaystyle u = x^2$

Then

$\displaystyle \frac{du}{dx} = 2x$

And then:

$\displaystyle du = 2x ~dx$

Which turns your integral into:

$\displaystyle \left[\int e^u ~du\right]_{u = x^2}$

Or you can recognize that:

$\displaystyle \frac{d}{dx} \left[e^{x^2}\right] = e^{x^2}2x$

Your integral would be:

$\displaystyle \int \frac{d}{dx}\left[e^{x^2}\right] ~dx$

Which is really easy as well.