Question Bank • QB_SUB_EM_0128
MSQ • 2 Marks
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
Which of the following are second-order linear Partial Differential Equations of parabolic type?
Options:
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(A) One-dimensional transient heat conduction equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
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(B) One-dimensional wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
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(C) Two-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
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(D) One-dimensional moisture diffusion equation $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$
Step-by-Step Solution & Explanation
For $A u_{xx} + B u_{xt} + C u_{tt} = 0$, discriminant $B^2 - 4AC = 0$ is parabolic. The heat and diffusion equations have $B = 0, C = 0 \implies B^2 - 4AC = 0$ (parabolic). Wave equation is hyperbolic ($B^2 - 4AC > 0$) and Laplace is elliptic ($B^2 - 4AC < 0$).
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