# Is there a simple solution to this (ordinary) differential equation?

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?

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Doubt it: wolframalpha.com/input/… – user7530 Feb 23 '13 at 3:59

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 $y(x)$.
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. – Minh Feb 23 '13 at 4:29