Question

The integrating factor of the differential equation $x\frac{dy}{dx}-y=2x^2$
(a) e^−x

(b) e^−y

(c) $\frac{1}{x}$

(d) x

Answer 1

$\left(\mathrm{c}\right)\frac{1}{x}$


We have,

$$\begin{gathered} x\frac{dy}{dx}-y=2x^2 \\ \Rightarrow \frac{dy}{dx}-\frac{1}{x}y=2x \\ \mathrm{Comparing}\mathrm{with}\frac{dy}{dx}+Py=Q,\mathrm{we}\mathrm{get} \\ P=-\frac{1}{x} \\ Q=2x \\ \mathrm{Now}, \\ \mathrm{I}.\mathrm{F}.=e^{-\int \frac{1}{x}dy} \\ =e^{-\log \left|x\right|} \\ =e^{\log \left|\frac{1}{x}\right|} \\ =\frac{1}{x} \end{gathered}$$