Question

The solution of the differential equation \( \frac{dy}{dx} = e^x + 1 \) is:

• A. \( y = e^x + C \)
• B. \( y = x + e^x + C \)
• C. \( y = xe^x + C \)
• D. \( y = x(e^x + 1) + C \)
• E. \( y = e^x + Cx \)

Hint:

When $dy/dx$ is purely a function of $x$, the solution is simply the anti-derivative of that function plus a constant.

Answer 1

The Correct Option is B

Concept: This is a simple first-order differential equation where the variables are separable. We can integrate the right-hand side directly with respect to \( x \).

Step 1: Separate the variables.
\[ dy = (e^x + 1) dx \]

Step 2: Integrate both sides.
\[ \int dy = \int (e^x + 1) dx \] \[ y = \int e^x dx + \int 1 dx \]

Step 3: Final evaluation.
\[ y = e^x + x + C \] This matches option (B).