In order to solve this kind of problem you would have to know some basic "differential equations". The equation you have here, ax= dvx/dt= d^2x/dt^2, is a "second order ordinary differential equation with constant coefficients". It can be shown that any solution to such an equation is "of the type" e^{at} for some constant a. The first derivative of that is ae^{at} and the second derivative is a^2e^{at}. Putting those into the equation, ax= d^2x/dt^2= a^2e^{at}= 100x= 100e^{at}. Dividing both sides of a^2e^{at}= 100e^{at} by e^{at} we have the "characteristic equation" a^2= 100 so a= 10i or a= 10i. That means the general solution is a linear combination of e^{10it} and e^{10it}.
But since this problem clearly involves only real numbers we would prefer to write the solution using only real numbers. We can do that by remembering that e^{ait}= cos(at)+ isin(at). That is, the solution to a "linear ordinary differential equation with constant coefficients", when the solutions to the characteristic equation are imaginary numbers can be written in terms of sine and cosine.
Of course the problem here did NOT ask you to find the solution it proposed a solution and asked you to show that it was correct. To do that you only had to use Calculus to show that it did satisfy the conditions.
