Given that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-2xy}{x^2 + y^2}$, use the substitution \

Question

Given that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-2xy}{x^2 + y^2}$, use the substitution $y = vx$, where $v$ is a function of $x$ to show that$$x \frac{\mathrm{d}v}{\mathrm{d}x} = -\frac{3v + v^3}{1 + v^2}.$$ Hence, show that the solution for the given differential equation is $3x^2 y + y^3 = c$ where $c$ is an arbitrary constant.

Accepted Answer

(a) $x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{3v+v^3}{1+v^2}$ (b) $3x^2y+y^3=c$, $c\in\mathbb R$

Homogeneous equations

Homogeneous equations

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