Answered: 24 24 The two-dimensional Laplace…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

The two-dimensional Laplace equation,
f(x,y) = 5x-9y-6
2²4 2²4
-= 0, describes potentials and steady-state temperature distributions in a plane. Show that the following function satisfies the two-dimensional Laplace equation.
dx

Expert Solution

Step 1

The two-dimensional Laplace equation is given by d^2f/dx^2 + d^2f/dy^2 = 0, where f(x, y) is a function of two independent variables x and y. This equation is used to describe the potentials and steady-state temperature distributions in a plane.

To show that the function f(x, y) = 5x - 9y - 6 satisfies the two-dimensional Laplace equation, we need to take the second partial derivatives of f with respect to x and y, and then add them together.

The first partial derivative of f with respect to x is given by ∂f/∂x = 5, and the second partial derivative of f with respect to x is given by ∂^2f/∂x^2 = 0. This is because the second derivative of a linear function with respect to its independent variable is always zero.