The solution of the differential equation \( e^{\frac{dy}{dx}}=x+1;\ y(0)=5 \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( y=(1+x)\log_{e}(1+x)-x+5 \)
Taking the logarithm of both sides,
\( \displaystyle \frac{dy}{dx}=\log_{e}(1+x) \)
Integrating by parts (with 1 as the second function),
\( \displaystyle y=x\log_{e}(1+x)-\int\frac{x}{1+x}\,dx \)
\( \displaystyle =x\log_{e}(1+x)-\int\left(1-\frac{1}{1+x}\right)dx \)
\( =x\log_{e}(1+x)-x+\log_{e}(1+x)+C \)
\( =(1+x)\log_{e}(1+x)-x+C \)
Using \( y(0)=5 \): \( 0-0+C=5 \), so C = 5 and
\( y=(1+x)\log_{e}(1+x)-x+5 \)
Shortcut: Options (2) and (3) give \( y(0)=0 \), and Option (1) has \( \frac{dy}{dx}=\log_{e}x \) (it is not even defined at x = 0). Only Option (4) satisfies both the equation and \( y(0)=5 \).