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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differential Equations

NDA 2018 · Mathematics · Differential Equations

What is the solution of the differential equation (d x)/(d...

What is the solution of the differential equation \frac{d x}{d y}=\frac{x+y+1}{x+y-1} ?

Options

  1. Ay−x+4ln⁡(x+y)=cy-x+4 \ln (x+y)=c
  2. By+x+cln⁡(x+y)=cy+x+c \ln (x+y)=c
  3. Cy−x+ln⁡(x+y)=cy-x+\ln (x+y)=c
  4. Dy+x+2ln⁡(x+y)=cy+x+2 \ln (x+y)=c

Correct answer

C. y−x+ln⁡(x+y)=cy-x+\ln (x+y)=c

Step-by-step solution

Let t=x+y⇒dtdy=dxdy+1t=x+y \Rightarrow \frac{d t}{d y}=\frac{d x}{d y}+1 So, dtdy−1=t+1t−1\frac{d t}{d y}-1=\frac{t+1}{t-1}⇒dtdy=t+1+t−1t−1=2tt−1⇒∫t−1tdt=2∫dy⇒t−log⁡t=2y+C1⇒x+y−log⁡(x+y)−2y=C1⇒y−x+log⁡(x+y)=−C1=C\begin{aligned} & \Rightarrow \quad \frac{d t}{d y}=\frac{t+1+t-1}{t-1}=\frac{2 t}{t-1} \\ & \Rightarrow \quad \int \frac{t-1}{t} d t=2 \int d y \\ & \Rightarrow \quad t-\log t=2 y+\mathrm{C}_1 \\ & \Rightarrow \quad x+y-\log (x+y)-2 y=\mathrm{C}_1 \\ & \Rightarrow y-x+\log (x+y)=-\mathrm{C}_1=\mathrm{C} \end{aligned}$$
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