2017 SRJC P1 Q6

Express \(\frac{5{{x}^{2}}-3}{x-{{x}^{3}}}\) in partial fractions.[2]

Using the substitution \(v=\frac{y}{x}\), \(x\ne 0\), show that the differential equation \(y\left( {{x}^{2}}+{{y}^{2}} \right)+x\left( 3{{x}^{2}}-5{{y}^{2}} \right)\frac{\mathrm{d}y}{\mathrm{d}x}=0\) can be reduced to \(\left( 5{{v}^{2}}-3 \right)\frac{\mathrm{d}v}{\mathrm{d}x}=4\left( \frac{v-{{v}^{3}}}{x} \right)\).

Hence, using the results in part (a), show that the general solution is given by \({{y}^{3}}\left( {{x}^{2}}-{{y}^{2}} \right)=Cx\) where \(C\) is an arbitrary constant.[7]