Solve the following differential equation: cosec x y dy dx +x^2y^… - Vidyadip

Question

Solve the following differential equation:
$\text{cosec }x\ \log\text{ y}\frac{\text{dy}}{\text{d}x}+x^2\text{y}^2=0$

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Answer

$\text{cosec }x.\ \log\text{y}\frac{\text{dy}}{\text{d}x}=-x^2\text{y}^2$$\Rightarrow\frac{\log\text{ y}}{\text{y}^2}\text{ dy}=-\text{x}^2\sin\text{x dx}$
Integrating both sideswe get
$\Rightarrow-\frac{\log\text{ y}}{\text{y}}\frac{1}{\text{y}}=-[-\text{x}^2\cos\text{x}+2\int\text{x}\cos\text{x dx}]$
$=-[-\text{x}^2\cos\text{x}+2(\text{x}\sin\text{x}-\int1.\sin\text{x dx}]$
$\therefore\frac{\log\text{y}}{\text{y}}-\frac{1}{\text{y}}=-\text{x}^2\cos\text{x}+2\text{x}\sin\text{x}+2\cos\text{x}+\text{c}$

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