Prove that for $x0$, the substitution $y=ux$ reduces the differential equation $\left( y-x \right)\

Question

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

Accepted Answer

(i) shown; (ii) $\frac12\ln\left(\frac{y^2}{x^2}+2\right)-\frac1{\sqrt2}\tan^{-1}\left(\frac{y}{\sqrt2x}\right)=x+C$

2020 RI P1 Q3

2020 RI P1 Q3

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