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Mathematics LibreTexts

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.