Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
Identify the following partial differential equations in the given order based on their generic form $A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu = G$:
Options:
(A) (i) Wave, (ii) Heat, (iii) Laplace
(B) (i) Heat, (ii) Laplace, (iii) Wave
(C) (i) Laplace, (ii) Wave, (iii) Heat
(D) (i) Heat, (ii) Wave, (iii) Laplace
Official Answer Key:
A
Step-by-Step Solution & Explanation
- (i) Second order in both time and space; discriminant B² − 4AC > 0 → hyperbolic → wave equation.
- (ii) First order in time, second order in space; B² − 4AC = 0 → parabolic → heat (diffusion) equation.
- (iii) ∇²u = 0, second order in both spatial variables with no time term; B² − 4AC < 0 → elliptic → Laplace equation.
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