GATE XE (Engineering Sciences) 2025 - Question 12

Section: Engineering Mathematics | Marks: 1

Consider the second order Partial Differential Equation (PDE) 4𝑥2 𝜕2𝑢 𝜕𝑥2 + 4(𝑥+ 𝑦) 𝜕2𝑢 𝜕𝑥𝜕𝑦+ (𝑥2 + 𝑦2) 𝜕2𝑢 𝜕𝑦2 − 𝑢= 0. Then which one of the following statements is correct ?

• A. The PDE is hyperbolic in the region {(𝑥, 𝑦) ∈ℝ2: −1 < 𝑥< 0, 𝑦< 0}
• B. The PDE is hyperbolic in the region {(𝑥, 𝑦) ∈ℝ2: 1 < 𝑥< ∞, 𝑦< 0}
• C. The PDE is elliptic in the region {(𝑥, 𝑦) ∈ℝ2: 0 < 𝑥< 1, 𝑦> 0}
• D. The PDE is parabolic in the region {(𝑥, 𝑦) ∈ℝ2: 1 < 𝑥< ∞, 𝑦< 0}

Answer: A. The PDE is hyperbolic in the region {(𝑥, 𝑦) ∈ℝ2: −1 < 𝑥< 0, 𝑦< 0}