Let us show that there is a unique solution, given that
Clearly is a solution with these initial conditions, so look at ; this satisfies . We want to show that for all x.
Multiplying by we see that for all x. Note that . Therefore, by the Mean Value Theorem,
Now a real sum of squares can only be zero if each summand is zero. So we have, as required, for all x.
[For those of an applied bent, we are just doing the classical thing of conserving the total energy.]
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