No, that's not correct. You tried to solve the problem with 3 variables. You integrated both sides with respect to x while treating v as the variable too. absurdity
You must substitute that for dy/dx into your differential equation then proceed via separation of variables.
I'm sorry to say that this is not correct, because you're integrating with respect to the wrong variable (v instead of x).
First apply the substitution that's given to you. . So now we need to rewrite :
. Since , we end up with
Thus, the ODE becomes . This is now a separable equation.
Can you proceed?
EDIT: Man, am I slow!!