Question

Solve the differential equation:
$x\frac{dy}{dx}+y=3x^2-2$

Answer 1

Given $x\frac{dy}{dx}+y=3x^2-2$
$\frac{dy}{dx}+\frac{1}{x}.y=3x-\frac{2}{x}......\left(i\right)$
Here $P=\frac{1}{x}$ and $Q=3x-\frac{2}{x}$
I.F $=e^{\int Pdx}=e^{\int \frac{1}{x}dx}=e^{\log x}=x$
Multiplying equation $\left(i\right)$ by I.F
$\frac{dy}{dx}+\frac{y}{x}.x=(3x-\frac{2}{x}).x$
$\frac{dy}{dx}+y=3x^2-2$
On integrating, we get
$xy=\int (3x^2-2)dx+c$
$xy=x^3-2x+c$ which is the required solution.