First Order Homogeneous Differential Equations

[drcameron]

Homework Statement

Find the general solution of the following homogeneous differential equations:

(i) $$\frac{du}{dx} = \frac{4u-2x}{u+x}$$
(ii) $$\frac{du}{dx} = \frac{xu+u^{2}}{x^{2}}$$

(You may express your solution as a function of u and x together)

Homework Equations

There are no relevant equations to this solution

The Attempt at a Solution

(i) $$\frac{du}{dx} = 4 - \frac{6x}{u+x}$$
I could then use the substition y=u+x with dy/dx = du/dx + 1 to give:
$$\frac{dy}{dx} = 5 - \frac{6x}{y}$$.
Now I'm really lost as shouldn't the y be on the top or am I missing something really stupid here?

(ii) Similar problem to above - should get it from (i) but a hint would go a long way.

 

[Dick]

The usual trick in the homogeneous case it to use the substitution y=u/x. Did you try that? It should make it separable.

 

[drcameron]

Many thanks, using a more appropriate substitution helps a lot. The second equation then just fell into place for me as a result.