2020 RI P1 Q3
(i)
Prove that for $x>0$, the substitution $y=ux$ reduces the differential equation $\left( y-x \right)\left( \frac{\text{d}y}{\text{d}x}-\frac{y}{x} \right)={{y}^{2}}+2{{x}^{2}}$ to
$\left( \frac{u}{{{u}^{2}}+2}-\frac{1}{{{u}^{2}}+2} \right)\left( \frac{\text{d}u}{\text{d}x} \right)=1$.
[2]
(ii)
Hence find the general solution of the differential equation
$\left( y-x \right)\left( \frac{\text{d}y}{\text{d}x}-\frac{y}{x} \right)={{y}^{2}}+2{{x}^{2}}$ for $x>0$.
[3]
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