Discussion

The partial differential equation obtained by eliminating arbitrary constants from

\[ z = a e^{bx} \sin by \]

is :

A.\(\frac{\partial^{2} z}{\partial y^{2}} + \frac{\partial z}{\partial x} = 0\)
B.\(\frac{\partial^{2} z}{\partial y^{2}} + \frac{\partial^{2} z}{\partial x^{2}}\) = 0
C.\(\frac{\partial^{2} z}{\partial y^{2}} + x\frac{\partial^{2} z}{\partial x^{2}}\) = 0
D.\(\frac{\partial^{2} z}{\partial y^{2}} + x\frac{\partial z}{\partial x} = 0\)

Answer: B

The equation is \(\dfrac{\partial^{2} z}{\partial y^{2}} + \dfrac{\partial^{2} z}{\partial x^{2}}\) = 0.

Differentiate \(z = a e^{bx}\sin by\):

\[ \begin{aligned} z_{x} &= b z, \quad z_{xx} = b^{2} z \\ z_{y} &= ab\, e^{bx}\cos by \\ z_{yy} &= -ab^{2} e^{bx}\sin by = -b^{2} z \end{aligned} \]

Adding, \(z_{xx} + z_{yy} = 0\): the constants \(a\) and \(b\) are eliminated, and \(z\) satisfies Laplace's equation.

Option A would need \(z_{x} = b^{2}z\), but \(z_{x} = bz\). Options C and D bring in a factor \(x\) that the differentiation never produces.

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