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Integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is
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Verified Answer
The correct answer is:
$e^{\left(\frac{1}{y}\right)}$
Given differential equation is
$$
\begin{array}{rlrl}
& & \left(x+2 y^3\right) \frac{d y}{d x}=y^2 \\
\Rightarrow & y^2 \frac{d x}{d y} & =x+2 y^3 \Rightarrow & \frac{d x}{d y}-\frac{x}{y^2}=2 y \\
\therefore & & I F & =e^{\int-\frac{1}{y^2} d y}=e^{\frac{1}{y}}
\end{array}
$$
$$
\begin{array}{rlrl}
& & \left(x+2 y^3\right) \frac{d y}{d x}=y^2 \\
\Rightarrow & y^2 \frac{d x}{d y} & =x+2 y^3 \Rightarrow & \frac{d x}{d y}-\frac{x}{y^2}=2 y \\
\therefore & & I F & =e^{\int-\frac{1}{y^2} d y}=e^{\frac{1}{y}}
\end{array}
$$
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