The integrating factor of left(x+2 y^3right) | vedclass

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Question

(A) The given differential equation is $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$.Rearranging the equation to the form $\frac{dx}{dy} + P(y)x = Q(y)$,

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$e^{\left(\frac{1}{y}\right)}$

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(A) The given differential equation is $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$.Rearranging the equation to the form $\frac{dx}{dy} + P(y)x = Q(y)$, we get:$\frac{dx}{dy} = \frac{x+2y^3}{y^2} = \frac{x}{y^2} + 2y$$\frac{dx}{dy} - \frac{1}{y^2}x = 2y$Here, $P(y) = -\frac{1}{y^2}$.The integrating factor $(IF)$ is given by $e^{\int P(y) dy}$.$IF = e^{\int -\frac{1}{y^2} dy} = e^{\int -y^{-2} dy} = e^{-(-y^{-1})} = e^{\frac{1}{y}}$.

The integrating factor of $\left(x+2 y^3\right) \frac{d y}{d x}=y^2$ is

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