How do you solve a hard differential equation with an integrating factor?

[anirudhreddy]

A Hard differential equation!

Solve:

dy/dx = (x^2) + y

 

[genneth]

The rules of this forum requires you to show some working, so that we know where to begin helping.

Can you solve the homogeneneous equation: dy/dy - y = 0 ?
Can you find a particular integral?

 

[HallsofIvy]

That is a first order linear differential equation with constant coefficients- actually, it's about the easiest you could come up with. genneth suggested solving the "homogeneous equation" first. That would work.

But for linear first order equations, there is a standard formula for the "integrating factor". You could also use that.

 

[mjsd]

relevant equation:
if \frac{dy(x)}{dx}+P(x)\,y(x) = Q(x)
then
y(x) = e^{-\int P(\eta)\,d\eta} \int Q(x)\;e^{\int P(\xi)\,d\xi}\,dx

if you understand this you probably understand how to do your problem

 

[anirudhreddy]

thx guys


so...

first i should write it in the form

dy/dx + (-1)y = (x^2)

is that right?

 

[mjsd]

the next step into better understanding this is to prove the formula above...

 

[sennyk]

Proof hint

The way I always proved this was to make the differential equation exact first. Then the rest is algebra; ahem, calculus.

 

[Jwink3101]

dy/dx-y=x^2 is a good start

To make your integrating factor, you do Exp(integral(-1dx)) (i hope that makes sense). Work it from there and see where you get.