Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& x^2{f}^{\prime }(x)-2xf(x)=3\\ & \left(\frac{x^2{f}^{\prime }(x)-2xf(x)}{{\left(x^2\right)}^2}\right)=\frac{3}{{\left(x^2\right)}^2}\\ & \Rightarrow \frac{d}{dx}\left(\frac{f(x)}{x^2}\right)=\frac{3}{x^4}\end{array}$ Integrating both sides $\begin{array}{rl}& \frac{f(x)}{x^2}=-\frac{1}{x^3}+C\\ & f(x)=-\frac{1}{x}+C{x}^2\\ & \text{ put }x=1\\ & 4=-1+C\Rightarrow C=5\\ & f(x)=-\frac{1}{x}+5x^2\\ & \text{ Now }2\times f(2)=2\times \left[-\frac{1}{2}+5\times 2^2\right]\\ & =39\end{array}$