Question

Find the general solution of the differential equation $x\frac{dy}{dx}+2y=x^2$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ x\frac{dy}{dx}+2y=x^2 \\ \Rightarrow \frac{dy}{dx}+\frac{2}{x}y=x.....\left(1\right) \\ \mathrm{Clearly},\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{linear}\mathrm{differential}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{form} \\ \frac{dy}{dx}+Py=Q \\ \mathrm{where}P=\frac{2}{x}\mathrm{and}Q=x. \\ \therefore \mathrm{I}.\mathrm{F}.=e^{\int Pdx} \\ =e^{\int \frac{2}{x}dx} \\ =e^{2\log x} \\ =x^2 \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{of}\left(1\right)\mathrm{by}\mathrm{I}.\mathrm{F}.=x^2,\mathrm{we}\mathrm{get} \\ {x}^2\left(\frac{dy}{dx}+\frac{2}{x}y\right)=x^{2}x \\ \Rightarrow x^2\frac{dy}{dx}+2xy=x^3 \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ {x}^{2}y=\int x^{3}dx+C \\ \Rightarrow x^{2}y=\frac{x^4}{4}+C \\ \Rightarrow y=\frac{x^2}{4}+C{x}^{-2} \\ \mathrm{Hence},y=\frac{x^2}{4}+C{x}^{-2}\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$