The number of solutions of the differential equation \( \displaystyle\frac{dy}{dx}-y=1 \), given that \( y(0)=1 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
The equation \( \frac{dy}{dx}-y=1 \) is linear, with integrating factor \( e^{-x} \):
\( \displaystyle \frac{d}{dx}\left(y\,e^{-x}\right)=e^{-x} \)
\( \displaystyle y\,e^{-x}=-e^{-x}+C \)
\( \displaystyle y=Ce^{x}-1 \)
The condition \( y(0)=1 \) fixes the constant:
\( \displaystyle 1=C-1\ \Rightarrow\ C=2 \)
So there is exactly one solution, \( y=2e^{x}-1 \).