What is \( \frac{d^2y}{dx^2} \left( \frac{dx}{dy} \right)^2 \) equal to ?
For the next two (02) items that follow : Consider the differential equation \( e^{x+y} \frac{dy}{dx} = e^{x-y} \)
- A. -2 ✓
- B. -1
- C. 0
- D. 2
Correct Answer: A. -2
Explanation
Rewriting the differential equation: \( e^x e^y \frac{dy}{dx} = e^x e^{-y} \implies \frac{dy}{dx} = e^{-2y} \). Differentiating again with respect to x gives \( \frac{d^2y}{dx^2} = -2e^{-2y} \frac{dy}{dx} = -2e^{-4y} \). The requested expression is \( -2e^{-4y} \times (e^{2y})^2 = -2 \).
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