Solve the following differential equation dy dx -x - Vidyadip

Question

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\log\text{x}$

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Answer

We have
$\frac{\text{dy}}{\text{dx}}-\text{x}\log\text{x}$
$\Rightarrow\text{dy}=(\text{x}\log\text{x})$
Integrating boh sides we get,
$\int\text{dy}=\int(\text{x}\log\text{x})\text{dx}$
$\Rightarrow\text{y}=\int\text{x}\times\log\text{x dx}$
$\Rightarrow\text{y}=\log\text{x}\int\text{x dx}-\int\Big[\frac{\text{d}}{\text{dx}}(\log\text{x})\int\text{x dx}\Big]\text{dx}$
$\Rightarrow\text{y}=\log\text{x}\times\frac{\text{x}^2}{2}-\int\Big(\frac{1}{\text{x}}\times\frac{\text{x}^2}{2}\Big)\text{dx}$
$\Rightarrow\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\int\frac{\text{x}}{2}\text{ dx}$
$\Rightarrow\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\frac{\text{x}^2}{4}+\text{C}$
hence, $\text{y}=\frac{1}{2}\text{x}^2\log\text{x}-\frac{\text{x}^2}{4}+\text{C}$ is the solutin to the given differential equation.

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