If I have a second order differential equation (like in SHM) ,ie, (d^2x)/(dt^2)=-kx, (1) the solution is of the form Acos(wt+B) (2). This is obviously true because by substituting i get the result. But the answer could very well have been [-(kx^3)/6 +AX^2+BX + C]. Why do we not take this answer to solve the SHM problems? Also, if I am solving the above differential equation, instead of equating (1) to -kx i could have equated it to zero and found the solution, and added to (2). It would still satisfy the equation. So I could keep adding many such solutions to (2). But then there would be an infinite number of solutions! How is that possible?? And finally what if I have (d^2x)/(dt^2)=-kx + C .What would be the general solution (I know that the frequency will not change).For example if I have a spring with two masses m1 and m2 on either side and I apply a force F1 and F2 respectivley on each of them, what would be the equation of motion of the center of mass? It would be very helpful if anyone can tell me the equation or how to solve the Differential equation.