**Hermite's Equation** of order *k* has the form

where

We know from the previous section that this equation will have series solutions which both converge and solve the differential equation everywhere.

Hermite's Equation is our first example of a differential equation, which has a polynomial solution.

As usual, the generic form of a power series is

We have to determine the right choice for the coefficients (

As in other techniques for solving differential equations, once we have a "guess" for the solutions, we plug it into the differential equation.
Recall that

and

Plugging this information into the differential equation we obtain:

or after rewriting slightly:

Next we shift the first summation up by two units:

Before we can combine the terms into one sum, we have to overcome another slight obstacle: the second summation starts at

Evaluate the 0th term for the second sum:
.
Consequently, we do not change the value of the second summation, if we start at *n*=0 instead of *n*=1:

Thus we can combine all three sums as follows:

Therefore our recurrence relations become:

After simplification, this becomes

Let us look at the special case, where

What about the odd coefficients?

and

What about

Since

Consequently, the solution has only 3 non-zero coefficients, and hence is a polynomial. This polynomial

(or a multiple of this polynomial) is called the

It turns out that the Hermite Equation of positive integer order *k* always has a polynomial solution of order *k*. We can even be more precise: If *k* is odd, the initial value problem
will have a polynomial solution, while for *k* even, the initial value problem
will have a polynomial solution.

Find the solution satisfying the initial conditions

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