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  • Explain solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 54 maths textbook solution.

Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 54 maths textbook solution.

Answers (1)

Answer : (c) y e^{x}+x^{2}=C

Hint : Convert the given equation in the form of  \frac{d y}{d x}+P(x) y=Q(x)

Given : e^{x} d y+\left(y e^{x}+2 x\right) d x=0

Explanation : e^{x} d y=-\left(y e^{x}+2 x\right) d x

\begin{aligned} &\Rightarrow \frac{e^{x} d y}{d x}=-y e^{x}-2 x \\ &\Rightarrow \frac{d y}{d x}=\frac{-y e^{x}-2 x}{e^{x}} \\ &\Rightarrow \frac{d y}{d x}=-y-\frac{2 x}{e^{x}} \\ &\Rightarrow \frac{d y}{d x}+y=-\frac{2 x}{e^{x}} \end{aligned}

So, the given equation is linear in y

I F=e^{\int d x}=e^{x}

Hence, the general solution

\begin{aligned} &\Rightarrow y e^{x}=\int-\frac{2 x}{e^{x}} e^{x} d x+C \\ &\Rightarrow y e^{x}=-2 \int x d x+C \\ &\Rightarrow y e^{x}=-\frac{2 x^{2}}{2}+C \end{aligned}

\begin{aligned} &\Rightarrow y e^{x}=-x^{2}+C \\ &\Rightarrow y e^{x}+x^{2}=C \end{aligned}

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