Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
A partial differential equation containing dependent variable $u$ is given by $A\frac{\partial^2 u}{\partial x^2} + 2B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} = 0$. The equation is parabolic if:
Options:
(A) Linear and homogeneous
(B) Non-linear and homogeneous
(C) Linear and non-homogeneous
(D) Non-linear and non-homogeneous
Official Answer Key:
C
Step-by-Step Solution & Explanation
A PDE is linear if the dependent variable u and all its partial derivatives appear only to the first power and are not multiplied together, with coefficients depending only on the independent variables (x, y) — exactly the case here (A, B, C, D, E, F are functions of x, y only, never of u). It is homogeneous only if every term contains u or a derivative of u (i.e., the equation equals zero with no standalone forcing term). Here G ≠ 0 is a standalone term independent of u, making the equation non-homogeneous. Answer: (C).
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