Question

The solution of the differential equation \(\frac{dy}{dx}=\frac{xy+y}{xy+x}\) is

• A. \(y-x=\log \left(\frac{x}{y}\right)+c\)
• B. \(x-y=\log \left(\frac{x}{y}\right)+c\)
• C. \(y+x=\log \left(\frac{x}{y}\right)+c\)
• D. \(y+x=\log \left(\frac{y}{x}\right)+c\)

Hint:

Factorise numerator and denominator to separate the variables as (y+1)/y dy = (x+1)/x dx.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Factorise: \(xy+y=y(x+1)\) and \(xy+x=x(y+1)\).

Step 2: Separate the variables
\(\frac{dy}{dx}=\frac{y(x+1)}{x(y+1)}\Rightarrow\frac{y+1}{y}dy=\frac{x+1}{x}dx\).

Step 3: Integrate
\(\int\left(1+\frac{1}{y}\right)dy=\int\left(1+\frac{1}{x}\right)dx\)
\(y+\log y=x+\log x+c\).

Step 4: Rearrange
\(y-x=\log x-\log y+c=\log\left(\frac{x}{y}\right)+c\).

Final Answer:
The solution is y - x = log(x/y) + c. \[ \boxed{\text{(A) }y-x=\log \left(\dfrac{x}{y}\right)+c} \]