I'm trying to solve the following differential equation:

$$ \frac{dy}{dx} = - \frac{3x + 2y}{2y}. $$

It looks pretty simple, yet it's not separable, linear, or exact. It is of the form

$$ \frac{dy}{dx} = f(y/x). $$

I could do the substitution $v = y/x$, and I know it would look pretty ugly, but is there a better or simpler method?


One way I can think is to solve: $$ \frac{dy}{dt} = -3 x - 2y \\ \frac{dx}{dt} = 2 y $$ say by matrix exponentiation. Then one can invert $ x(t) $ to find $ y(x) $.

  • $\begingroup$ For the qualitative end behavior of the solution, I like the matrix solution. The associated matrix is $$\begin{pmatrix} 0 & 2 \\ -3 & -2 \end{pmatrix},$$ which leads to the eigenvalues $-1 \pm i\sqrt{5}$. So I can have overall sense of the family of solutions. In particular, I can tell that they must "go" through the origin. $\endgroup$ – Minh Feb 23 '13 at 4:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.