Solve the following differential equations:(xy^2+2x)dx+(x^2y+2y)d… - Vidyadip

Question

Solve the following differential equations:$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y+2y})\text{dy}=0$

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Answer

We have,$(\text{xy}^2+2\text{x})\text{dx}+(\text{x}^2\text{y}+2\text{y})\text{dy}=0$
$\Rightarrow\text{x(y}^2+2)\text{dx+y}(\text{x}^2+2)\text{dy}=0$
$\Rightarrow\text{x(y}^2+2)\text{dx}=-\text{y}(\text{x}^2+2)\text{dy}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+2)}\text{dx}=-\frac{\text{y}}{(\text{y}^2+2)}\text{dy}$
Integration both sides, we get
$\int\frac{\text{x}}{\text{x}^2+2}\text{dx}=-\int\frac{\text{y}}{\text{y}^2+2}\text{dy}$
$\Rightarrow\frac{1}{2}\int\frac{2\text{x}}{\text{x}^2+2}\text{dx}=-\frac{1}{2}\frac{2\text{y}}{\text{y}^2+2}\text{dy}$
$\Rightarrow\frac{1}{2}\log|\text{x}^2+2|=-\frac{1}{2}\log|\text{y}^2+2|+\log\text{C}$
$\Rightarrow\frac{1}{2}\log|\text{x}^2+2|+\frac{1}{2}\log|\text{y}^2+2|=\log\text{C}$
$\Rightarrow\log|\text{x}^2+2|+\log|\text{y}^2+2|=2\log\text{C}$
$\Rightarrow\log\big(|\text{x}^2+2||\text{y}^2+2|\big)=\log\text{C}^2$
$\Rightarrow\big(|\text{x}^2+2||\text{y}^2+2|\big)=\text{C}^2$
$\Rightarrow(\text{x}^2+2)(\text{y}^2+2)=\text{K}$
$\Rightarrow\text{y}^2+2=\frac{\text{K}}{\text{x}^2+2}$

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