Question

If \( \dfrac{d^2y}{dx^2} = \sin x + e^x \), \( y(0) = 3 \) and \( \left.\dfrac{dy}{dx}\right|_{x=0} = 4 \), then the equation of the curve is

• A. \( y = 4 + 2x + e^x - \sin x \)
• B. \( y = 2 + 3x + e^x - \sin x \)
• C. \( y = 2 + 4x + e^x - \sin x \)
• D. \( y = 4 + 2x + e^x + \sin x \)

Hint:

When initial conditions are given, always apply them immediately after integration to find constants.

Answer 1

The Correct Option is C

Step 1: Integrate the second derivative.
\[ \frac{d^2y}{dx^2} = \sin x + e^x \] \[ \frac{dy}{dx} = -\cos x + e^x + C_1 \] Step 2: Use the given condition on \( \dfrac{dy}{dx} \).
At \( x = 0 \), \[ 4 = -1 + 1 + C_1 \Rightarrow C_1 = 4 \] Step 3: Integrate again.
\[ y = -\sin x + e^x + 4x + C_2 \] Step 4: Use the condition \( y(0) = 3 \).
\[ 3 = 0 + 1 + 0 + C_2 \Rightarrow C_2 = 2 \] Step 5: Write the equation of the curve.
\[ y = 2 + 4x + e^x - \sin x \] Step 6: Conclusion.
The required equation of the curve is \( y = 2 + 4x + e^x - \sin x \).