Question

Löse \(\Delta \varphi=0\) mit dem Produktansatz \(\varphi(x, y, z)=X(x) \cdot Y(y) \cdot Z(z) \neq 0\) (Lösungen evtl. komplex.)

Short answer

The solution to the equation \(\Delta \varphi=0\) using the given product approach is found by separating the variables, rearranging and solving the resulting differential equations for each variable.

Step-by-step solution

Analyze and Apply the Product Solution Approach

Apply the product solution approach to \(\Delta \varphi=0\), which gives us \(\varphi(x, y, z)=X(x) \cdot Y(y) \cdot Z(z)\). Split each variable, it gives \(X''(x)Y(y)Z(z) + X(x)Y''(y)Z(z) + X(x)Y(y)Z''(z)=0\)

Variable Separation

Divide both sides by \(\varphi(x, y, z)=X(x) \cdot Y(y) \cdot Z(z)\) to separate the variables. We get \( \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} + \frac{Z''(z)}{Z(z)}=0 \)

Re-arrange for Each Variable

Rewrite the equation as follows: \( \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)}= -\frac{Z''(z)}{Z(z)} = k \), where \(k\) is the separation constant. We then obtain the three differential equations, \(X''(x)-kX=0, Y''(y)-kY=0, Z''(z)+kZ=0\)

Solve Each Separated Differential Equation

Solve each separated differential equation to find the solutions for \(X(x)\), \(Y(y)\), and \(Z(z)\). This requires solving second order differential equations for each variable separately with the given conditions.

Key concepts

Separation of Variables

Separation of variables is a vital technique used in solving partial differential equations, such as Laplace's equation: \(\Delta \varphi = 0\). This method involves simplifying a complex equation by breaking it down into simpler, one-variable components. By assuming that the solution can be expressed as the product of functions, each depending on a single variable, we can make the process more manageable.

In our current scenario, we start with the assumption: \(\varphi(x, y, z) = X(x) \cdot Y(y) \cdot Z(z)\). This assumption allows us to rewrite the given equation so that we can treat each variable independently.

• Divide by the assumed product \(X(x) \cdot Y(y) \cdot Z(z)\), resulting in the separated form \(\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} + \frac{Z''(z)}{Z(z)} = 0\).
• Introduce a constant \(k\) to facilitate further separation, ultimately leading to individual ordinary differential equations for each component \(X(x)\), \(Y(y)\), and \(Z(z)\).

Through this method, a seemingly unsolvable three-variable problem becomes a set of simpler equations, each of which can be solved separately based on standard techniques.

Partial Differential Equations

Partial differential equations (PDEs) involve functions with multiple variables and their partial derivatives. These equations describe a wide range of phenomena, from physics to engineering. An example of a PDE is Laplace's equation which looks like \(\Delta \varphi = 0\). In our exercise, this equation models states of equilibrium, for instance the electric potential in a charge-free region.

PDEs like Laplace's equation are often challenging to solve directly due to their multivariate nature. However, by using certain methods such as separation of variables, we can transform them into simpler ordinary differential equations (ODEs).

• PDEs typically need specific boundary conditions to be fully solved.
• They are categorized as linear, semilinear, or nonlinear based on the linearity of their terms.

Having a good grasp of PDEs is crucial for advanced studies in mathematics, physics, and engineering disciplines. They are the backbone of many theoretical and practical applications.

Product Solution Method

The product solution method is a powerful approach in solving specific types of partial differential equations, particularly those that can be separated into simpler one-variable equations. This method leverages the assumption that the solution can be written as a product of functions, each dependent on only one variable, such as \(\varphi(x, y, z) = X(x) \cdot Y(y) \cdot Z(z)\).

By applying the product solution method, we can decompose a complex multivariable problem into smaller, more manageable parts.

• This method helps in converting a PDE into ordinary differential equations (ODEs), which are easier to handle.
• Each function is then solved separately, often accompanied by initial or boundary conditions specific to the problem.
• The collective solutions of these ODEs provide a comprehensive solution to the original PDE.

This method not only simplifies the solving process but also uncovers the underlying structure of the solution, giving insight into how the components of the problem interact.

Complex Solutions

When solving partial differential equations, like Laplace's equation, we often consider both real and complex solutions, especially in physics and engineering where waveforms and oscillations are involved. Complex solutions add an important dimension to these problems by accounting for both amplitude and phase.

In the context of our product solution, considering complex solutions means admitting terms of the form \(e^{i\theta}\), where \(i\) is the imaginary unit. This form, thanks to Euler's formula, can also be expressed in terms of sine and cosine functions, making it highly flexible for modeling periodic behavior.

• Complex solutions are valuable in cases involving harmonic oscillations and other cyclical phenomena.
• They often simplify mathematical expressions and can make certain integral and differential operations easier.

Overall, looking beyond real solutions allows for a more complete understanding of the system being modeled and can lead to more elegant mathematical formulations.