It is given that $2xy\frac{\mathrm{d}y}{\mathrm{d}x}=x^2-y^2$ where $x0$, \(y

Question

It is given that $2xy\frac{\mathrm{d}y}{\mathrm{d}x}=x^2-y^2$ where $x>0$, $y<0$. Using the substitution $u=\frac{y}{x}$, show that the differential equation can be transformed to $\frac{2u}{1-3u^2}\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{1}{x}$. Hence find the general solution of $y$ in terms of $x$.[6]

Accepted Answer

$y=-x\sqrt{\dfrac13\left(1-\dfrac{A}{x^3}\right)}$, where $A$ is an arbitrary constant

2025 DHS P1 Q1

2025 DHS P1 Q1

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