5.1 Cookbook
Let me start with a recipe that describes
the approach to separation of variables, as exemplified
in the following sections, and in later chapters. Try to trace
the steps for all the examples you encounter in this
course.
- Take care that the boundaries are
naturally described in your variables (i.e., at the boundary
one of the coordinates is constant)!
- Write the unknown function as a product
of functions in each variable.
- Divide by the function, so as to have a
ratio of functions in one variable equal to a ratio of
functions in the other variable.
- Since these two are equal they must both
equal to a constant.
- Separate the boundary and initial
conditions. Those that are zero can be re-expressed as
conditions on one of the unknown functions.
- Solve the equation for that function
where most boundary information is known.
- This usually determines a discrete set of
separation parameters.
- Solve the remaining equation for each
parameter.
- Use the superposition principle (true for
homogeneous and linear equations) to add all these solutions
with an unknown constants multiplying each of the
solutions.
- Determine the constants from the
remaining boundary and initial conditions.