Question

Find the integrating factor of $\frac{dy}{dx}+\frac{y}{x}=x^2$.

Answer 1

Suppose we have the first order differential equation

$\frac{dy}{dx}+Py=Q$

where $P$ and $Q$ are functions involving x only.

We multiply both sides of the differential equation by the integrating factor $I$ which is defined as

$e^{\int Pdx}$

Here $P=\frac{1}{x}$

Hence, the integrating factor $I$ $=e^{\int \frac{1}{x}dx}=e^{\ln x}=x$