First we find the general solution. If the third derivative of f is 0, the second derivative of f must be a constant: f ^{(2)}(x) = A. The first derivative must be a linear function with slope A: f '(x) = Ax + B. The original function must look like this: Now we use the initial conditions to fill in the constants. When x = 0 we want f(0) = 0, so which means C = 0. When x = 0 we want f ' (0) = 1, so 1 = A(0) + B which means B = 1. When x = 0 we want f "(0) = 4, so 4 = A. Putting everything we know about the constants together, we get Of course, we can check our answer by taking 3 derivatives of f to make sure that f ^{(3)}(x) = 0 (it does) and making sure that f, f ', and f " have the correct values when x = 0. |