An integrating factor of the equation $\left(1+y+x^2 y\right) d x+\left(x+x^3\right) d y=0$ is

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Aicharya · Accepted Answer

Given, differential equation is, $ \begin{aligned} & d x\left(1+y+x^2 yight)+\left(x+x^3ight) d y=0 \ \Rightarrow \quad & \frac{d y}{d x}=-\left[\frac{1+y+x^2 y}{x+x^3}ight] \end{aligned} $ $ \begin{aligned} & \Rightarrow \quad \frac{d y}{d x}=-\left\{\frac{1}{x\left(1+x^2ight)}+\frac{\left(1+x^2ight) y}{x\left(1+x^2ight)}ight\} \ & \Rightarrow \quad \frac{d y}{d x}-\frac{y}{x}=-\frac{1}{x\left(1+x^2ight)} \ & \end{aligned} $ Here, $p=-\frac{1}{2}, Q=-\frac{1}{x\left(1+x^2ight)}$ $ \begin{aligned} 	ext { Integrating factor } & \equiv e^{\int p d x} \ & =e^{\int\left(-\frac{1}{x}ight)} d x \ & =e^{-\log x} \ & =e^{\log \left(\frac{1}{x}ight)} \ & =\frac{1}{x} \end{aligned} $

An integrating factor of the equation $\left(1+y+x^2 y\right) d x+\left(x+x^3\right) d y=0$ is

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