Question

The integrating factor of the differential equation $\left(x^{2}-1 \frac{dy}{dx}+2\right)ky = x$ is

• A. $\frac{1}{x^{2}-1}$
• B. $x^{2}-1$
• C. $\frac{x^{2}-1}{x}$
• D. $\frac{x}{x^{2}-1}$

Answer 1

Answer: (B)

Given differential equation is
$\left(x^{2}-1\right) \frac{dy}{dx}+2xy = x$
$\Rightarrow \quad \frac{dy}{dx}+\frac{2x}{x^{2}-1}.y = \frac{x}{x^{2}-1}$
This is in linear form.
Integrating factor $= \int\limits_{e} \frac{2x}{x^{2}-1}dx = \int\limits_{e} \frac{dt}{t}$
where $t=x^{2} -1$
$= e^{log\,t}=x^{2} - 1$
Hence, required integrating factor $=x^{2} -1.$