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. |