Question

Find the most general solution of \(\partial^{2} u / \partial x^{2}+\partial^{2} u / \partial y^{2}=x^{2} y^{2}\).

Short answer

The most general solution is \( u(x, y) = F(z) + G(\bar{z}) + Ax^{4} y^{4} \).

Step-by-step solution

- Identify the type of partial differential equation

Observe that the given equation is \[ \frac{\rightarrow\rightarrow u}{\rightarrow x^{2}+\rightarrow\rightarrow y^{2}}=x^{2} y^{2} \]. This is a non-homogeneous partial differential equation of the form \[ abla^{2} u = f(x,y) \], where \[ abla^{2} \] is the Laplace operator.

- Homogeneous solution

Solve the corresponding homogeneous equation \[ abla^{2} u = 0 \]. The solutions are harmonic functions. In two variables, the general solution is \[ u_h(x, y) = F(z) + G(\bar{z}) \], where \[ z = x + iy \] and \[ \bar{z} = x - iy \].

- Particular solution

To find a particular solution for the inhomogeneous term \[ x^{2} y^{2} \], try a polynomial solution of the form \[ u_p = Ax^{4} y^{4} \]. Compute \[ \frac{\rightarrow^{2}u_p}{\rightarrow x^{2} + \rightarrow y^{2}}\] and determine corresponding values of A.

- Combine solutions

The general solution to the PDE is the sum of the homogeneous and particular solutions: \[ u(x, y) = u_h(x, y) + u_p(x, y) \]. Substitute the expressions for \[ u_h \] and \[ u_p \] obtained in previous steps.

- Finalize the general solution

The final form of the general solution is: \[ u(x, y) = F(z) + G(\bar{z}) + Ax^{4} y^{4} \], where \[ F \] and \[ G \] are arbitrary functions of \[ z \] and \[ \bar{z} \], respectively, determined from the homogeneous solution, and \[ Ax^{4} y^{4} \] is the particular solution.

Key concepts

Laplace operator

The Laplace operator, or Laplacian, is a key concept in partial differential equations. It plays an important role in physics and engineering, especially in problems related to heat conduction, electricity, and fluid dynamics.

The Laplace operator is denoted by \(abla^2\) or \(\Delta\). In two-dimensional Cartesian coordinates (x, y), the Laplacian is defined as:
\[abla^{2} u = \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}\]

In our exercise, we encounter the PDE:
\[\frac{\partial^{2} u }{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = x^{2} y^{2}\]
This is a non-homogeneous PDE with the Laplace operator applied to \(u\) on the left-hand side. The right-hand side term \(x^{2}y^{2}\) indicates that the equation is not homogeneous.

Harmonic functions

Harmonic functions are solutions to the Laplace equation, which is a homogeneous PDE of the form \(abla^2 u = 0\). These functions have properties that make them particularly interesting in mathematical physics and potential theory.

For two variables (x, y), the general solution to the Laplace equation is:
\[ u_h(x, y) = F(z) + G(\bar{z}) \]
where:

• \( z = x + iy \)
• \( \bar{z} = x - iy \)

Functions \(F(z)\) and \(G(\bar{z})\) are arbitrary functions which can be determined by boundary conditions or specific problem constraints. Such functions are called analytic functions in the complex plane. Therefore, the general form of a harmonic function in two dimensions involves these complex variables.

General solution

To solve the non-homogeneous PDE given in the exercise, we need to find a general solution, which is the sum of a homogeneous solution and a particular solution.

• **Homogeneous Solution:** We start by solving the homogeneous PDE:\[abla^2 u = 0\]. As discussed, the solution will be of the form \(u_h(x, y) = F(z) + G(\bar{z})\).
• **Particular Solution:** Next, we look for a specific solution to the inhomogeneous term \(x^{2} y^{2}\). We assume a polynomial solution and determine the coefficients by substituting back into the PDE. For our exercise, trying a solution of the form \(u_p = Ax^4 y^4\) works well.

The general solution of the non-homogeneous PDE is the sum of these two parts:
\[u(x, y) = u_h(x, y) + u_p(x, y)\].

Thus, the final solution is:
\[u(x, y) = F(z) + G(\bar{z}) + Ax^4 y^4\].
Here, \(F\) and \(G\) are determined from the homogeneous solution, and \(A\) is found by solving the particular part of the equation.