/*  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 the RK4 method.  */

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

#define N 1024
static double t0=0,u0=1,tN=10,h;

double f(double t,double u){
    return exp(-u)*sin(t+u);
}
double rk4(double t,double u){
    double k1=h*f(t,u);
    double k2=h*f(t+h/2,u+k1/2);
    double k3=h*f(t+h/2,u+k2/2);
    double k4=h*f(t+h,u+k3);
    return u+(k1+2*(k2+k3)+k4)/6;
}

int main(){
    printf("This is problem 1 on exam part 2.\n");
    h=(tN-t0)/N;
    double u=u0;
    for(int n=0;n<N;n++){
        u=rk4(t0+n*h,u);
    }
    printf("u(%g)=%.14e\n",tN,u);
    if(fabs(u-2.34310412817669e-01)<1e-10){
        printf("Congratulations!  Your answer is correct!\n");
    } else {
        printf("Please try again.  Your answer is incorrect.\n");
    }
    return 0;
}