Understanding Differential Equations: Solving for y' with u substitution

[Axial]

So I've just started Differential equations and already I am stuck.

I have the Question:

xy' = y² + y It also gives me (u = y/x) and i can see how to get u.So now i have y' =(y²/x) + u Now I am stuck, I'm not really sure of the exact way of going about this.

 

[Mark44]

This equation is separable, so that you can get dy and the y terms on one side, and dx and the x term on the other. Then integrate, not forgetting the constant of integration.

 

[HallsofIvy]

If you want to treat it as a "homogeneous" equation, which is what the "hint", to take u= y/x implies, then dividing the equation by x gives y'= y^2/x+ y/x as you say and, yes, that is y'= y^2/x+ u. But you can also write y^2/x= (y/x)y= uy. Your equation becomes y'= uy+ u= u(y+1) which is again separable:
dy/(y+1)= u du.

I tend to agree with Mark44, that, since it is already separable, doing it as dy/(y^2+ y)= dx/x seems more obvious.

 

[Axial]

I would have done it automaticly like that, exept it states * use u=y/x *

This is what the given solution is:

I think i might stick with my answer and see what he says.

 

[Mark44]

Check your solution by calculating dy/dx. It should be that for your solution xdy/dx = y² + y. If so, you're in business. If not, you need to go back and see where you went wrong.