Question

The equation of the curve through \((1,0)\), whose slope is \(\frac{y-1}{x^2+x}\), is:

• A. \(2x(y-1)+x+1=0\)
• B. \((x+1)(y-1)+2x=0\)
• C. \(x(y-1)(x+1)+2=0\)
• D. \(x(y+1)+y(x+1)=0\)

Hint:

Use partial fractions for rational integrals.

Answer 1

The Correct Option is B

Concept: \[ \frac{dy}{dx} = \frac{y-1}{x(x+1)} \]

Step 1: Separate variables.
\[ \frac{dy}{y-1} = \frac{dx}{x(x+1)} \]

Step 2: Integrate.
\[ \ln|y-1| = \ln\left|\frac{x}{x+1}\right| + C \]

Step 3: Apply condition \((1,0)\).
\[ \Rightarrow (x+1)(y-1)+2x=0 \]