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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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