Question

Solve the differential equation: $(1+x^2)\frac{dy}{dx}+2xy=4x^2$.

Answer 1

Differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$
$\frac{dy}{dx}+\frac{2x}{1+x^2}y=\frac{4x^2}{1+x^2}$
It is a linear differential equation
$\frac{dy}{dx}+Py=Q$ where $P=\frac{2x}{1+x^2}$
$Q=\frac{4x^2}{1+x^2}$
I.F. $=e^{\int Pdx}$
I. F. $=e^{\int \frac{2x}{1+x^2}dx}$
$1+x^2=t$
$2xdx=dt$

I.F.$=e^{\int \frac{dt}{t}}$
I.F.$=e^{{\log }_{e}t}$
I.F.$=t=(1+x^2)$
$y(I.F)=\int (I.F.)Qdx+c$
$y(1+x^2)=\int \frac{4x^2}{1+x^2}(1+x^2)dx+c$
$y(1+x^2)=\int 4x^{2}dx+c$
$y(1+x^2)=4\int x^{2}dx+c$
$y(1+x^2)=4\left(\frac{x^3}{3}\right)+c$
$y(1+x^2)=\frac{4}{3}\left(x^3\right)+c$.