Suppose that you actually got $\displaystyle \ln y+\ln(v^2+1)=k,$ then put $\displaystyle k=\ln c$ ('cause both of them are constants), thus your equation becomes $\displaystyle y\big(v^2+1\big)=c,$ because of injectivity.
the ODE is not exact, so you should find an "intergrating factor" to multiply with the original ODE so that the NEW ODE is solvable. R u interested on how to solve the problem?