Find the solution of the differential equation \[\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 + y^2}{

Question

Find the solution of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^2 + y^2}{xy}$$ $$(x + y) \frac{\mathrm{d}y}{\mathrm{d}x} = x - y$$ $$x(x + y) \frac{\mathrm{d}y}{\mathrm{d}x} + y(3x + y) = 0$$

Accepted Answer

(a) $y^2=x^2(\ln(x^2)+A)$, on real branches with $xy\ne0$. (b) $x^2-2xy-y^2=A$, including $y=(-1\pm\sqrt2)x$. (c) $2x^2y^2+4x^3y=A$, including $y=0,-2x$. In (b)–(c), exclude nonzero points with $y=-x$; constants are independent.

Homogeneous equations

Homogeneous equations

Solution:

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