In my differential equations course one of the problems asked the following:
Show that $sin(t^3)$ cannot be a solution of any equation of the form $$z''' + a_1(t)z'' + a_2(t)z' + a_3(t)z = 0$$ where a1(t), a2(t), and a3(t), are continuous over an open interval containing 0.
The solution was essentially that $z(t) = 0$ was unique by the basic existence and uniqueness theorem, therefore making it so $z=sin(t^3)$ cannot be a solution. However I don't see why the functions could be reversed to argue the opposite.
My professor's explanation was that "$z(t)=0$ is clearly a solution". How can this be answered more rigorously?