Solve the following differential equation xy(y+1)dy=(x^2+1)dx - Vidyadip

Question

Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$

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Answer

We have $\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$$\Rightarrow\{\text{y}(\text{y}+1)\}\text{dy}=\frac{\text{x}^2+1}{\text{x}}\ \text{dx}$
$\Rightarrow(\text{y}^2+\text{y})\text{dy}=\Big(\text{x}+\frac{1}{\text{x}}\Big)\text{dx}$
Integrating both sides, we get
$\int(\text{y}^2+\text{y})\text{dy}=\int\Big(\text{x}+\frac{1}{\text{x}}\Big)\text{dx}$ $=\int\text{y}^2\text{dy}+\int\text{y dy}=\int\text{x dx}+\int\frac{1}{\text{x}}\text{ dx}$ $\Rightarrow\frac{\text{y}^3}{3}+\frac{\text{y}^3}{2}=\frac{\text{x}^2}{2}+\log|\text{x}|+\text{C}$ Hence, $\frac{\text{y}^3}{3}+\frac{\text{y}^3}{2}=\frac{\text{x}^2}{2}+\log|\text{x}|+\text{C}$ is the required solution.

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