Q55 / 90 · JEE Main 2012 (26 May Online) · Mathematics
The integrating factor of the differential equation $\left(x^2-1 \frac{d y}{d x}+2\right) x y=x$ is
- A$\frac{1}{x^2-1}$
- Correct answerB$x^2-1$
- C$\frac{x^2-1}{x}$
- D$\frac{x}{x^2-1}$
Solution
Given differential equation is
$$
\begin{aligned}
& \left(x^2-1\right) \frac{d y}{d x}+2 x y=x \\
& \Rightarrow \quad \frac{d y}{d x}+\frac{2 x}{x^2-1} \cdot y=\frac{x}{x^2-1}
\end{aligned}
$$
This is in linear form.
$$
\begin{aligned}
& \text { Integrating factor }=\int \frac{2 x}{x^2-1} d x=\int_e \frac{d t}{t} \\
& \text { where } \begin{aligned}
& =x^2-1 \\
& =e^{\log t}=x^2-1
\end{aligned}
\end{aligned}
$$
Hence, required integrating factor $=x^2-1$.
Hence, the correct answer is option: 'B'.