Solve the following differential equation:(y + 3x^ 2 ) dx dy =x. - Vidyadip

Question

Solve the following differential equation:$\text{(y + 3x}^{2})\frac{\text{dx}}{\text{dy}}=\text{x}$.

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Answer

Given equation can be written as $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=\text{3x}^{2}$ OR $\frac{\text{dy}}{\text{dx}}-\frac{1}{\text{x}}\cdot\text{y}=\text{3x}$ $\text{I.F.}=\text{e}^{-\int\frac{1}{\text{x}}\text{dx}}=\text{e}^{-\log\text{x}}=\text{e}^{\log\frac{1}{\text{x}}}=\frac{1}{\text{x}}$ $\therefore\text{ solution is, y}\cdot\frac{1}{ \text{x}}=\int\text{3x}\cdot\frac{1}{\text{x}}\text{dx}=\text{3x + c}$$\Rightarrow\text{y}=\text{3x}^{2}+\text{cx}$.

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