Question

General solution of differential equation \(\rm \frac {dy}{dx} + y = 1, (y \ne 1)\), is:
1. \(\rm \log \left| \frac {1}{1 - y} \right| = x + C\)
2. log |1 - y| = x + C
3. log |1 + y| = x + C
4. \(\rm \log \left| \frac {1}{1 - y} \right| = -x + C\)
5. None of these

Answer 1

Correct Answer - Option 1 : \(\rm \log \left| \frac {1}{1 - y} \right| = x + C\)

Calculation:

The given differential equation \(\rm \frac {dy}{dx} + y = 1, (y \ne 1)\) is in the variable separable form.

\(\Rightarrow \rm \frac {dy}{dx} = 1 - y\)

Separating the variables, we get:

⇒ \(\rm \frac{dy}{1-y}=dx\)

On integrating, we get:

⇒ -log (1 - y) = x + C

⇒ log (1 - y)^-1 = x + C                  [m log n = log n^m]

⇒ \(\rm \log \left| \frac {1}{1 - y} \right| = x + C\).