Solve the following differential equation x dy dx + =0 - Vidyadip

Question

Solve the following differential equation
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0$

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Answer

We have,$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}-\cot\text{y}$
$\Rightarrow\frac{1}{\text{x}}\ \text{dx}=-\frac{1}{\cot\text{y}}\ \text{dy}$
$\Rightarrow\frac{1}{\text{x}}\ \text{dx}=-\tan\text{y dy}$
Integrating both sides, we get
$\int\frac{1}{\text{x}}\ \text{dx}=-\int\tan\text{y dy}$
$\Rightarrow\int|\text{x}|=-\int|\sec\text{y}|+\int\text{C}$
$\Rightarrow\int|\text{x}|=-\int|\cos\text{y}|+\int\text{C}$
$\Rightarrow\text{x}=\text{C}\cos\text{y}$
Hence, $\text{x}=\text{C}\cos\text{y}$ is the required solution.

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