Question

General solution of the differential equation $\frac{dy}{dx}=\frac{[x+y+1]}{[x+y-1]}$ is given by

• A. $x+y=\log |x+y|+c$
• B. $x–y=\log |x+y|+c$
• C. $y=x+\log |x+y|+c$
• D. $y=x\log |x+y|+c$

Answer 1

The correct option is C

$y=x+\log |x+y|+c$

Explanation for the correct option:

Step 1. Find the general solution:

$\frac{dy}{dx}=\frac{x+y+1}{x+y-1}$

Put $x+y=z$

Step 2. Differentiate it with respect to $x$

$1+\frac{dy}{dx}=\frac{dz}{dx}$

$\Rightarrow$ $\frac{dz}{dx}=1+\frac{z+1}{z-1}$/.;

$\Rightarrow$ $\frac{dz}{dx}=\frac{z+1+z-1}{z-1}$

$\Rightarrow$ $\frac{dz}{dx}=\frac{2z}{z-1}$

$\Rightarrow$$\frac{(z-1)}{z}dz=2dx$

$\Rightarrow$$dz-\frac{dz}{z}=2dx$

Step 3. By Integrating it, we get

$\int dz-\int \frac{dz}{z}=2\int dx$

$\Rightarrow z-\log \left|z\right|=2x+C$

Step 4. Put the value of $z$

$x+y-\log |\left(x+y\right)|=2x+C$

$\Rightarrow y=\mathbf{}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathbf{|}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{|}\mathbf{}\mathbf{+}\mathbf{}\mathit{C}$

Hence, Option ‘C’ is Correct.