What is the solution of the equation $xyp^2 + (3x^2 - 2y^2)p - 6xy=0$, where $p = \frac{dx}{dy}$

For $xy\ne0$, using Vieta $$p^2+\left(\frac{3x}y-\frac{2y}x\right)p+6=0,$$

factors as

$$\left(p-\frac{3x}y\right)\left(p-\frac{2y}x\right)=0$$

which is easy.

$y=0$ is also a solution.


Using quadratic formula, we have $$p=\frac{2y^2 - 3x^2\pm\sqrt{(3x^2-2y^2)^2+24x^2y^2}}{2xy}$$ $$=\frac{2y^2 - 3x^2\pm(3x^2+2y^2)}{2xy}. $$

That is: $$\frac{dy}{dx}=p=\frac{2y}{x} \text{ or } \frac{-3x}{y}. $$ In both cases, we can easily solve the ODE.

Tags:

Ordinary Differential Equations

Nonlinear System

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