%Differential Equation: dy/dx=y-(1/2)*exp(x/2).*sin(5*x)+5*exp(x/2).*cos(5*x); y(0)=0;
f=@(x,y) y-(1/2)*exp(x/2).*sin(5*x)+5*exp(x/2).*cos(5*x);
f1=@(x,y) exp(x/2).*sin(5*x);   %Analytic Solution
xinit=0; yinit=0;           %y(xinit)=yinit {y(0)=0}

xmax=10;
x=xinit:0.1:xmax;
h=x(2)-x(1);
m=length(x);
y=zeros(1,m);
y(1)=yinit;

for n=1:m-1
    k1=h*f(x(n),y(n));
    k2=h*f(x(n)+h,y(n)+k1);
    y(n+1)=y(n)+1/2*(k1+k2);
end;

yy=exp(x/2).*sin(5*x);      %Analytic Solution
hold on;
h=plot(x,y,'ro--','LineWidth',1.5);
plot(x,yy,'b-','LineWidth',2.5);
title('2nd Order RK Method','fontsize',14,'fontweight','bold');
grid on;
legend('2nd Order RK Method','Analytic','Location','NorthWest');
hold off;

Published with MATLAB® 7.10