Consider the Hermite Equation of order 5:
Find the solution satisfying the initial conditions a0=1, a1=0.
The solution will not be a polynomial.
Since a1=0, all odd coefficients will be zero.
Let's compute a few of the even coefficients:
From this it is not too hard to come up with the general formula:
This leads to the formula
for the solution to the given initial value problem.
The solution converges and solves the equation for all real numbers.
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