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FAS1998
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- I've attached an image of a solved problem. Can somebody explain the steps in the yellow box? I don't understand how they got to that point from the previous steps.
That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?phyzguy said:I suspect it is the first step that is bothering you.
[tex] \sum_{n=1}^{\infty}a_n \sin(\frac{n \pi x}{2}) \sinh(\frac{n \pi}{2}) = \sin(\frac{\pi x}{2}) [/tex]
This is true for all values of x. The only way this can be true is for all of the [itex] a_n[/itex] to be zero except [itex]a_1[/itex]. Is this the step that is troubling you? After this, the rest follows pretty easily.
FAS1998 said:That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?
The Laplace equation is a partial differential equation (PDE) that describes the behavior of steady-state systems. It is important in separation of variables because it allows us to break down a complex PDE into simpler, independent equations that can be solved individually.
Separation of variables involves assuming a solution to the Laplace equation in the form of a product of two functions, one depending only on the spatial variables and the other depending only on the temporal variable. This allows us to separate the PDE into two ordinary differential equations, which can be solved using standard techniques.
The boundary conditions for the Laplace equation in separation of variables are typically specified at the boundaries of the system. These can include Dirichlet boundary conditions, which specify the values of the solution at the boundaries, or Neumann boundary conditions, which specify the values of the derivative of the solution at the boundaries.
Yes, separation of variables can also be used for other types of PDEs, such as the heat equation and the wave equation. However, it is most commonly used for the Laplace equation due to its simple form and the availability of analytical solutions.
Yes, separation of variables can only be used for PDEs with certain boundary conditions, such as those that are linear and homogeneous. It also requires the PDE to have a simple form, which may not always be the case for more complex systems. In these cases, numerical methods may be more suitable for solving the PDE.