Question

If \[ y = 500 e^{7x} + 600 e^{-7x}, \quad \text{then show that} \quad \frac{d^2 y}{dx^2} = 49y. \] 

Hint:

When differentiating exponential functions, apply the chain rule carefully. Notice that \( \frac{d}{dx} e^{kx} = k e^{kx} \).

Answer 1

Step 1: Differentiate \( y = 500 e^{7x} + 600 e^{-7x} \) with respect to \( x \): \[ \frac{dy}{dx} = 500 \cdot 7 e^{7x} - 600 \cdot 7 e^{-7x} = 3500 e^{7x} - 4200 e^{-7x}. \] 

Step 2: Now, differentiate again to find the second derivative: \[ \frac{d^2 y}{dx^2} = 3500 \cdot 7 e^{7x} + 4200 \cdot 7 e^{-7x} = 24500 e^{7x} + 29400 e^{-7x}. \] 

Step 3: Factor out 49 from both terms: \[ \frac{d^2 y}{dx^2} = 49 \left( 500 e^{7x} + 600 e^{-7x} \right) = 49y. \] Thus, we have shown that \( \frac{d^2 y}{dx^2} = 49y \).