/*  Approximate a solution to the initial value problem

        u' = f(t,u) with u(t0) = u0

    on the interval [t0,tN] using N time steps of Taylor's second
    order method.

    Note:  If you change f you must also change ft and fu below.  */

#include <stdio.h>
#include <math.h>

#define N 1024
static double t0=1,u0=2,tN=6,h;

double f(double t,double u){
    /* Please fill in this function */
}
double ft(double t,double u){
    /* Please fill in this function */
}
double fu(double t,double u){
    /* Please fill in this function */
}
double tstep(double t,double u){
    double du=f(t,u);
    return u+h*(du+h/2*(ft(t,u)+fu(t,u)*du));
}

int main(){
    printf("This is problem 1 on quiz 2.\n");
    h=(tN-t0)/N;
    double u=u0;
    for(int n=0;n<N;n++){
        u=tstep(t0+n*h,u);
    }
    printf("u(%g)=%.14e\n",tN,u);
    if(fabs(u-6.16948945991662e-01)<1e-11){
        printf("Congratulations!  Your answer is correct!\n");
    } else {
        printf("Please try again.  Your answer is incorrect.\n");
    }
    return 0;
}