Discussion
Identify the following partial differential equations in the given order based on their generic form where \(c\) is a constant, and \(u_{1}\) and \(u_{2}\) are functions of \((x, t)\) and \((x, y)\), respectively:
(i) \(\frac{\partial^2 u_1}{\partial t^2} = c^2 \frac{\partial^2 u_1}{\partial x^2}\) (ii) \(\frac{\partial u_1}{\partial t} = c^2 \frac{\partial^2 u_1}{\partial x^2}\) (iii) \(\frac{\partial^2 u_2}{\partial x^2} + \frac{\partial^2 u_2}{\partial y^2} = 0\)
Answer: A
The order is (i) Wave equation, (ii) Heat equation, (iii) Laplace equation.
Equation (i) has a second derivative in time equal to \(c^{2}\) times a second derivative in space: \(u_{tt} = c^{2}u_{xx}\) is the one-dimensional wave equation, with \(c\) the wave speed. Equation (ii) has only a first derivative in time: \(u_{t} = c^{2}u_{xx}\) is the one-dimensional heat (diffusion) equation, with \(c^{2}\) the diffusivity. Equation (iii), \(u_{xx} + u_{yy} = 0\), has no time variable at all and is the two-dimensional Laplace equation for steady potentials.
The other options swap these labels. The quick test is the order of the time derivative: second order means waves, first order means diffusion, and none means a steady (Laplace) problem.
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