Solve the following differential equation dy dx - 2x= 3x - Vidyadip

Question

Solve the following differential equation
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$

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Answer

We have,
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
$\Rightarrow\text{dy}=\frac{\cos3\text{x}+\cos2\text{x}}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=\frac{4\cos^2\text{x}-3\cos\text{x}+2\cos^2\text{x}-1}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=(4\cos^2\text{x}-3+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{dy}[2(2\cos^2\text{x}-1)-1+2\cos\text{x}-\sec\text{x}]\text{dx}$
$\Rightarrow\text{dy}(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{ dx}$
Integrating both sides, we get
$\int\text{dy}=\int(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$
hence, $\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$ is the solution to the given differential equation.

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