Question

The equation of the curve passing through the origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ is :

Answer 1

Given that,$(1+x^2)\frac{dy}{dx}+2xy=4x^2$
Dividing both sides by $(1+x^2)$
$\Rightarrow \frac{dy}{dx}+\frac{2x}{1+x^2}.y=\frac{4x^2}{x^2}$
Which is a linear differential equation.
On comparing it with $\frac{dy}{dx}+Py=Q$, we get
$P=\frac{2x}{1+x^2},Q=\frac{4x^2}{1+x^2}$
$IF=e^{\int pdx}=e^{\int \frac{2x}{1+x^2}}$
Put $1+x^2=t\Rightarrow 2xdx=dt$
$IF=1+x^2=e^{\int \frac{dt}{t}}=e^{\log t}=e^{\log (1-x^2)}$
The general solution is
$.y.(1+x^2)=\int \frac{4x^2}{1+x^2}dx+C$
$\Rightarrow y.(1+x^2)=\int 4x^{2}dx+C$
$\Rightarrow y.(1+x^2)=4\frac{x^3}{3}+C...\left(i\right)$
Since, the curve passes through origin, then substituting
$x=0$ and $y=0$ in Eq. $\left(i\right)$, we get
$C=0$
The required equation of curve is
$y(1+x^2)=4\frac{x^3}{3}$
$\Rightarrow y=\frac{4x^3}{3(1+x^2)}$