I'm struggling solving this separable differential equations: (du/dv) = ((uv+u)/(uv+v)) I multiplied both sides by dv to get on the left side just du and cancel out dv, but after that don't know where to go. I've tried integration by parts but it doesn't look right! Any help would be greatly appreciated!!

To solve the separable differential equation du/dv = ((uv+u)/(uv+v)), you're on the right track by multiplying both sides by dv. This gives you du = ((uv+u)/(uv+v)) dv.

Now, to separate the variables, you want to rearrange the equation so that all terms involving du are on one side and all terms involving dv are on the other side.

Start by multiplying both sides by (uv+v) to get rid of the denominator:

du(uv+v) = (uv+u)dv.

Expand both sides:

du * uv + du * v = uv * dv + u * dv.

Rearrange the terms to group du and dv on one side:

du * uv - uv * dv = u * dv - du * v.

Factor out du on the left side and dv on the right side:

du(uv - v) = dv(u - v).

Now, divide both sides by (uv - v) to isolate du:

du = (dv(u - v))/(uv - v).

And divide both sides by (u - v) to isolate dv:

dv = (du(uv - v))/(u - v).

Now, you have separated the variables. You can integrate both sides to solve for u and v. The integrals will depend on the form of (uv - v)/(u - v), which can be simplified further.

I hope this helps! Let me know if you have any further questions.