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

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Aicharya · Accepted Answer

Given differential equation is $ \begin{aligned} & \left(x^2-1ight) \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} & 	ext { Integrating factor }=\int \frac{2 x}{x^2-1} d x=\int_e \frac{d t}{t} \ & 	ext { where } \begin{aligned} & =x^2-1 \ & =e^{\log t}=x^2-1 \end{aligned} \end{aligned} $ Hence, required integrating factor $=x^2-1$

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

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