Solving an ODE by the method of Integrating Factors

[1missing]

1. y' + y = x y^2/32. The problem states we need to solve this ODE by using the method of integrating factors. Every example I found on the internet involving this method was of the form:

y' + Py = Q

Where P and Q are functions of x only. In the problem I was given however, Q is a function of both x and y. If I try to proceed with the method I end up with:

(d/dx) e^x y = x e^x y^2/3

Annnnd that's where I'm stuck. Am I missing something here or was this problem incorrectly assigned?

 

[BvU]

So what do you have to do to make the right-hand side a function in $x$ only ?

 

[1missing]

Divide by y^2/3 on both sides of the original equation, but then I've got a term in front of the y' and that doesn't seem to get me closer to a solution.

 

[Chestermiller]

What is $\frac{d(y^{1/3})}{dx}$ equal to?

 

[1missing]

Son of a...I probably would've stared at this thing for a couple of days before I noticed that. That should get me to a solution, let me give it a go.

 

[vela]

I end up with:
(d/dx) e^x y = x e^x y^2/3

Annnnd that's where I'm stuck. Am I missing something here or was this problem incorrectly assigned?

You could also proceed by noting that the righthand side is equal to $x e^{x/3} (e^x y)^{2/3}$.