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