Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation is $$\begin{array}{rl}& \left(1+x^2\right)\frac{dy}{dx}+2xy=4x^2\\ & \Rightarrow \frac{dy}{dx}+\left(\frac{2x}{1+x^2}\right)y=\frac{4x^2}{1+x^2}\end{array}$$ This is linear diff. equation $$\mathrm{I}.\mathrm{F}=e^{\int \frac{2x}{1+x^2}dx}=e^{\log \left(1+x^2\right)}=1+x^2$$ Solution is $$\begin{array}{rl}& y\left(1+x^2\right)=\int \frac{4x^2}{1+x^2}\times 1+x^2+\mathrm{C}\\ & \Rightarrow y\left(1+x^2\right)=\frac{4x^3}{3}+\mathrm{C}\\ & \Rightarrow \text{ Required curve is }\\ & 3y\left(1+x^2\right)=4x^3(\because \mathrm{C}=0)\end{array}$$