Question

If \[ y = 3e^{5x} + 5e^{3x}, \quad \text{then} \quad \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} = \]

• A. \( -10y \)
• B. \( 15y \)
• C. \( -15y \)
• D. \( 10y \)

Hint:

For higher derivatives of exponential functions, differentiate each term separately and combine the results. Then, perform the necessary operations as required by the problem.

Answer 1

The Correct Option is C

Step 1: Find the first derivative.
We are given \( y = 3e^{5x} + 5e^{3x} \). First, compute the first derivative \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = 15e^{5x} + 15e^{3x}. \]
Step 2: Find the second derivative.
Next, compute the second derivative \( \frac{d^2 y}{dx^2} \): \[ \frac{d^2 y}{dx^2} = 75e^{5x} + 45e^{3x}. \]
Step 3: Subtract \( 8 \frac{dy}{dx} \).
Now, subtract \( 8 \frac{dy}{dx} \) from \( \frac{d^2 y}{dx^2} \): \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} = (75e^{5x} + 45e^{3x}) - 8(15e^{5x} + 15e^{3x}) = -15(3e^{5x} + 5e^{3x}) = -15y. \]
Step 4: Conclusion.
Thus, the correct answer is option (C), \( -15y \).