Solve the following differential equation dy dx -xe^x-5 2 + ^2x - Vidyadip

Question

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$

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Answer

$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$
$\text{dy}=\Big(\text{xe}^\text{x}-\frac{5}{2}+\cos^2\text{x}\Big)\text{dx}$
$\int\text{dy}=\int\text{xe}^\text{x}\text{dx}-\frac{5}{2}\int\text{dx}+\cos^2\text{x dx}$
$\int\text{dy}=\int\text{xe}^\text{x}\text{dx}-\frac{5}{2}\int\text{dx}+\int\Big(\frac{1+\cos2\text{x}}{2}\Big)\text{dx}$
$=\int\text{xe}^\text{x}-\frac{5}{2}\int\text{dx}+\frac{1}{2}\int\text{dx}+\frac{1}{2}\int\cos2\text{x dx}$
$\int\text{dy}=\int\text{xe}^\text{x}-2\int\text{dx}+\frac{1}{2}\int\cos2\text{x dx}$
$\text{y}=[\text{x}\times\int\text{e}^\text{x}\text{dx}-\int(1\times\int\text{e}^\text{x}\text{dx})\text{dx}]-2\text{x}+\frac{1}{2}\frac{\sin2\text{x}}{2}+\text{C}$
Using integration by parts
$\text{y}=\text{xe}^\text{x}-\text{e}^\text{x}-2\text{x}+\frac{1}{4}\sin2\text{x}+\text{C}$
$\text{y}=\text{xe}^\text{x}-\text{e}^\text{x}-2\text{x}+\frac{1}{4}\sin2\text{x}+\text{C}$

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