What diff. eq. have the prop. that the product of any 2 sols is also a sol?

[worldly sluice]

I just learned that for a system of linear diff. eq. then the set of solutions is a vector space (the sum of 2 sols is also a sol and the scalar multiple of a sol is also a sol) and this is avector space with + and scalar *, but is there a diff. eq such that the set of sols is a vector space with the operatios:

1. Addition in the vector space will be multiplication (so, any 2 sols then their product is also a sol)
2. scalar exponentiation ( so for a sol phi then phi^alpha for any real alpha is also a solution)

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@worldly sluice

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