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A Hermite Polynomial: Instructor's Guide
Main Ideas
- Solving differential equations with power series
- Using a recurrence relation to identify even and odd solution behavior
Students' Task
Estimated Time: 25 minutes
Prerequisite Knowledge
Students should be given the Hermite equation in simple form (after all variable transformations) and should be familiar with the power series method for solving differential equations.
Props/Equipment
- Tabletop Whiteboard with markers
Activity: Introduction
See the activities for 
Change the Independent Variable and Change the Dependent Variable for lead-ins to this activity.
Activity: Student Conversations
Students who have previously solved differential equations using power series should have relatively little trouble getting to a recurrence relation. Major sticking points are reindexing sums and/or writing out several terms from each sum to identify the coefficient of each power. Some students may have difficulty with identifying each such coefficient vanishing independently - for more detail, see SGA Series Solutions. Students should be encouraged to discuss what they know about the first coefficients in the power series (i.e., c0 and c1). In particular, they should be guided to recognize that these two coefficients are always arbitrary because the differential equation is second order, leading to the two independent solutions. They should then recognize that for this recurrence relation, it is useful to choose either c0 or c1 to be equal to zero, as this results in solutions in only even and odd powers.
Activity: Wrap-up
Some students may not recognize that the termination of one of the solutions is a feature of the recurrence relation that always arises for certain values of the constant, what these values are, or what they mean physically. This is an excellent topic for the wrap-up (once the class has arrived at a collective solution) as it helps form the bridge between the mathematics and the physics that students will shortly be studying. Emphasize that while the constant is one that may be arbitrary in general, the need for polynomial solutions will force us to choose only those values of the constant that result in solutions matching the relevant physics.