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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:… – user7530 Feb 23 '13 at 3:59
up vote 4 down vote accepted

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) $.

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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

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