Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& 10\frac{\mathrm{d}}{\mathrm{dx}}{\int }_1^{\mathrm{x}}\mathrm{f}(\mathrm{t})\mathrm{dt}=\frac{\mathrm{d}}{\mathrm{dx}}\left(5\mathrm{xf}(\mathrm{x})-{\mathrm{x}}^5-9\right)\\ & \Rightarrow 10f(x)=5f(x)+5{\mathrm{xf}}^{\prime }(\mathrm{x})-5{\mathrm{x}}^4\\ & \Rightarrow \mathrm{f}(\mathrm{x})+{\mathrm{x}}^4={\mathrm{xf}}^{\prime }(\mathrm{x})\\ & \Rightarrow \mathrm{y}+{\mathrm{x}}^4=\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\\ & \Rightarrow \frac{dy}{dx}+y\left(-\frac{1}{x}\right)=x^3\\ & \Rightarrow y{e}^{\int -\frac{1}{x}dx}=\int x^3{e}^{\int -\frac{1}{x}dx}+c\\ & \Rightarrow {\mathrm{ye}}^{-\mathrm{tn}|x|}=\int {\mathrm{x}}^3{\mathrm{e}}^{-\mathrm{tr}|x|}+\mathrm{c}\\ & \Rightarrow \frac{\mathrm{y}}{|\mathrm{x}|}=\int \frac{{\mathrm{x}}^3}{|\mathrm{x}|}+\mathrm{c}\\ & \Rightarrow \frac{\mathrm{y}}{\mathrm{x}}=\int {\mathrm{x}}^2+\mathrm{c}\\ & \Rightarrow \frac{\mathrm{y}}{\mathrm{x}}=\frac{{\mathrm{x}}^3}{3}+\mathrm{c}\end{array}$
Put $x=1$ in given equation
$\begin{array}{rl}& \Rightarrow 0=5\mathrm{f}(1)-1-9\Rightarrow \mathrm{f}(1)=2\\ & \Rightarrow \frac{2}{1}=\frac{1}{3}+\mathrm{c}\Rightarrow \mathrm{c}=\frac{5}{3}\\ & \Rightarrow \frac{\mathrm{f}(3)}{3}=\frac{27}{3}+\frac{5}{3}\\ & \Rightarrow \mathrm{f}(3)=32\end{array}$