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Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 7

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Answer:  y=\left(\frac{x-1}{x}\right) e^{x}+\frac{C}{x}

Hint: To solve this equation we will use differentiate different.

Give: x \frac{d y}{d x}+y=x e^{x}

Solution:  \frac{d y}{d x}+P y=Q

\begin{aligned} &P=\frac{1}{x^{\prime}} f=e^{x} \\ & \end{aligned}

I\! f=e^{\int P d x}

\begin{aligned} &=e^{\int \frac{1}{x} d x} \\ & \end{aligned}

=e^{\log e^{x}} \\

=x \\

y I\! f=\int f I\! f

\begin{aligned} &y x=\int e^{x} x \\ & \end{aligned}

y x=x e^{x}+\int e^{x}+C \\

y x=(x+1) e^{x}+C \\

y=y \frac{(x-1)}{x} e^{x}+\frac{C}{x}

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