[Math] Find the general solution to the differential eq:$ (x^2 + y^2) + xyy’=0$

ordinary differential equations

Find the general solution to the differential equation:

$$(x^2 + y^2) + xyy'=0.$$

Hmm, I'm not sure what method to use to solve this but it looks like I can solve it with separable method?

Best Answer

Dividing by $xy$, $$\frac x y + \frac y x = -y'$$

This motivates the substitution $u = y/x$, so that

$$y = xu \implies y' = u + xu'$$

Substituting into the equation then gives

$$\frac 1 u + u = -u - xu'$$

and simplifying leads to

$$xu' = -\frac{1 + 2u^2}{u}$$

Can you finish from here?

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