Separation of variables possible in this problem?

In summary, the equation can be separable if a change of variables is made to take into account linear combinations of x and y.
  • #1
DuckAmuck
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TL;DR Summary
Second order linear partial differential equation
Is it possible to use separation of variables on this equation?
[tex] au_{xx} + bu_{yy} + c u_{xy} = u + k[/tex]
Where u is a function of x and y, abck are constant.
I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed.

Thank you.
 
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  • #2
Yes, something more clever will be required because presence of the ##u_{xy}## term as well as the nonhomogeneous term ##k## mean that straightforward separation of variables will not work. Getting rid of ##k## is easy: let ##u(x,y) = f(x,y) - k## and the equation for ##f## is homogeneous. Dealing with the ##u_{xy}## term isn't so easy, but sometimes a trick applies. First try separation of variables so you have an equation with terms such as ## \frac{X^{\prime\prime}(x)}{X(x)}##, etc. Then, differentiate that equation with respect to ##y##; the resulting equation might be separable.

Of course, even if the equation is separable you still need separable boundary conditions along curves of constant ##x## and ##y## in order for separation of variables to apply to your problem.

jason
 
  • #3
By looking at the discriminant ([itex] C^2 - 4AB [/itex] in this case) you can make a variable substitution and get a separable equation.
 
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  • #4
Yes, after making a change of variables.

The left hand side is [tex]
\begin{pmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{pmatrix}
\begin{pmatrix} a & \frac12 b \\ \frac12 b & c \end{pmatrix}
\begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{pmatrix} u
[/tex] This can be diagonalized to [tex]
\lambda_1 \frac{\partial^2 u}{\partial s^2} + \lambda_2 \frac{\partial^2 u}{\partial t^2}[/tex] where [itex]s[/itex] and [itex]t[/itex] are linear combinations of [itex]x[/itex] and [itex]y[/itex]. Setting [itex]\phi = u + k[/itex] now yields [tex]
\lambda_1 \frac{\partial^2 \phi}{\partial s^2} + \lambda_2 \frac{\partial^2 \phi}{\partial t^2} = \phi[/tex] which is separable.
 
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Related to Separation of variables possible in this problem?

1. Is separation of variables always possible in every problem?

No, separation of variables is not always possible in every problem. It depends on the type of equation and the variables involved. Some equations may not have a separable form, making it impossible to use separation of variables.

2. How do I know if a problem can be solved using separation of variables?

A problem can be solved using separation of variables if it is a partial differential equation with two or more variables that can be separated into two or more ordinary differential equations.

3. What is the general procedure for solving a problem using separation of variables?

The general procedure for solving a problem using separation of variables involves isolating the variables on one side of the equation, setting each variable equal to a constant, and then solving the resulting ordinary differential equations. The constants can then be solved for using initial or boundary conditions.

4. Can separation of variables be used for nonlinear equations?

Yes, separation of variables can be used for nonlinear equations as long as they can be separated into two or more ordinary differential equations. However, the resulting equations may be more difficult to solve compared to linear equations.

5. What are the advantages of using separation of variables to solve a problem?

Separation of variables can simplify the solution process for certain types of equations, making it easier to solve complex problems. It also allows for the use of techniques such as Fourier series and Laplace transforms, which can be useful in many scientific applications.

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