How do I do this?

Challenge: prove the following Let f(t) be a twice continuously differentiable function on R. Sup- pose that f"(t) satisfies the equation f " (t ) + c2f (t ) = 0, then there exist constants a and b such that f(t) = a cos(ct) + bsin(ct). Note: Upon proving this problem, you have successfully solved the first differential equation that you come across. It's important to note that solving a differential equation is very different than solving an equation in Algebra. To solve a differential equation, you must find all possible solutions and that requires justification. Most math students do not get to study this rigorously, through no fault of their own, and they see Differential Equation as just a recipe course, which is very far from the truth.Hint: Consider the following two functions: g(t) = f(t) cos( ct) - - f' (t) sin(ct) C h(t) = f(t) sin( ct) + - f' (t) cos( ct). Compute the derivative of h(t) and g(t) (they should come out very nice after you simplify). The derivative will say something about h(t) and g(t). Then use algebra to get the desired result. Now if you wonder how in the world did someone came up with those two functions, then welcome to the world of beautiful mathematics. Creativity is mesmerizing... and be- ing able to witness creativity is priceless