The multiplicatively advanced differential equations (MADEs) of form f(n)(t)=αf(βt) with α≠0, β>1 are studied along with a class of their solutions of type fμ,λ(t) defined on [0,∞). For λ∈Q+,μ∈R, the solutions fμ,λ(t) are extended to (−∞,∞) in a non-unique manner to obtain Schwartz wavelet solutions Fμ,λ(t) of the original MADE, with all moments of Fμ,λ(t) vanishing. Examples are studied in detail. The Fourier transform of each Fμ,λ(t) is computed and, in a number of examples, is related to the Jacobi theta function. Additional conditions sufficient for the uniqueness of certain MADE initial value problems are given. Conditions for decay and non-decay at −∞ are obtained. Decay rates at ±∞ in terms of familiar functions are established.