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The answer is positive: in general (apart of some artificial examples) if an IVP for an ODE $$ \dot x=f(t,x),\quad x(t)\in \mathbb R^n $$ have non unique solution it implies that there are uncountably many solutions. In these general setting this is a very nontrivial theorem which can be found in Hartman's book (Kneser's theorem). If, however, you are dealing with an ODE $$ \dot x=f(t,x),\quad x(0)=x_0, $$ where $x(t)$ is one-dimensional, it is a good (and simple) exercise to prove that if there are two solutions to this problem then there are infinitely (uncountably) many solutions.