Question

Which of the following satisfied the Laplace's equation in the plane
1. x^2 + y^2
2. x^2 - y^2
3. x + y
4. x - y

Answer 1

Answer on Question #79369 – Math – Differential Equations

Question

Which of the following satisfied the Laplace's equation in the plane

1. $x^{2} + y^{2}$

2. $x^{2} - y^{2}$

3. $x + y$

4. $x - y$

Solution

The two-dimensional (planar) Laplace's equation has the following form:

$$\frac {\partial^ {2} u}{\partial x ^ {2}} + \frac {\partial^ {2} u}{\partial y ^ {2}} = 0.$$

Substituting the given solutions into (1), we obtain:

$$u (x, y) = x ^ {2} + y ^ {2} \rightarrow 2 + 2 = 4 \neq 0,$$ $$u (x, y) = x ^ {2} - y ^ {2} \rightarrow 2 - 2 = 0,$$ $$u (x, y) = x + y \rightarrow 0 + 0 = 0,$$ $$u (x, y) = x - y \rightarrow 0 - 0 = 0.$$

Hence the functions $x^{2} - y^{2}, x + y$ and $x - y$ satisfy the Laplace's equation in the plane.

Answer: 2. $x^{2} - y^{2}$; 3. $x + y$; 4. $x - y$.

Answer provided by https://www.AssignmentExpert.com