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Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 27 maths textbook solution.

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Answer : (d)(x+C) e^{x+y}+1=0

Hint: Substitute x+y=t and separate x and t terms.

Given : \frac{d y}{d x}+1=e^{x+y}                   .....(i)

Explanation : Let y+x=t

Differentiate with respect to x

\Rightarrow \frac{d y}{d x}+1=\frac{d t}{d x}

Put in (i) we get

\begin{aligned} &\Rightarrow \frac{d t}{d x}=e^{t} \\ &\Rightarrow \frac{d t}{e^{t}}=d x \\ &\Rightarrow e^{-t} d t=d x \end{aligned}

Integrate both sides

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

\begin{aligned} &\Rightarrow-1=(x+C) e^{x+y} \\ &\Rightarrow(x+C) e^{x+y}+1=0 \end{aligned}


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