Ordinary differential equation and first integral - help!

Question

Ok, so I started learning ODE, and got my first H.W., but I have no idea even how to begin!

The question is to find the "first integral" of the following ODE:

$3t·(\cos t)u^2u'+(\cos t -t\sin t)u^3=0$

Later, I need to find "the first integral" of the equation that I get from comparing the previous "first integral" to a const. And if possible, to solve this ODE.

Of course, this is only the first ODE for this question, but I have no idea how to start solving it..

Answer 1

As a first strategy in such situations, try if you can identify functions $f$ and $g$ such that your ODE reads as $$ f(t)g'(u)u'+f'(t)g(u)=0 $$ since then you can employ the product rule to find that your equation is the same as $$ (f(t)g(u))'=0 $$ which has an easy first integral.

Answer 2

by using the separable method

$$\frac{du}{u}=\frac{t\sin t-\cos t}{3t\cos t}dt$$ $$\frac{du}{u}=(\frac{1}{3}\tan t-\frac{1}{3t})dt$$