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  • Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 36 subquestion (viii)

Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 36 subquestion (viii)

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Answer:  \begin{aligned} & x+y-1=C e^{-y} \\ & \end{aligned}

Give:  (x+y) \frac{d y}{d x}=1

Hint: Using integration by parts.

Explanation:  \begin{aligned} &(x+y) \frac{d y}{d x}=1 \\ & \end{aligned}

=\frac{d x}{d y}=x+y \\

=\frac{d x}{d y}-x=y 

This is a first order linear differential equation of the form

 \begin{aligned} &\frac{d y}{d x}+P y=Q \\ &P=-1 \text { and } Q=y \end{aligned}

The integrating factor  If of this differential equation is

 \begin{aligned} &I f=e^{\int P d y}\\ &=e^{\int-1 d y}\\ &=e^{-\int d y}\\ &=e^{-y} \quad\left[\int d y=y+C\right] \end{aligned}

Hence, the solution of different equation is

 \begin{aligned} &y I f=\int Q I f d x+C \\ &=y e^{-y}=\int y e^{-y} d y+C \ldots(i) \end{aligned}

Using integration by parts we have

\begin{aligned} &=\int y e^{-y} d y=\frac{y e^{-y}}{-1}-\int(1) e^{-y} d y+C \\ &=-y e^{-y}-\frac{e^{-y}}{-1}+C \\ &=-y e^{-y}+e^{-y}+C \end{aligned} 

 

From i

 =x e^{-y}=-y e^{-y}+e^{-y}+C

Divide by e^{-y}

 \begin{aligned} &=x=-y+1+\frac{C}{e^{-y}} \\ &=x=-y+1+C e^{-y} \end{aligned}

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