y''=xy, y(0)=0, y'(0)=1, (Airy equation) step size h=0.05 and interval is [-10,2], this is system of ode, i can already solve runge kutta method but also i need airy solution i can try but not make correct graph please help me...

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jamila nizamani on 14 Sep 2014

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https://nl.mathworks.com/matlabcentral/answers/154798-y-xy-y-0-0-y-0-1-airy-equation-step-size-h-0-05-and-interval-is-10-2-this-is-system-of

Edited: per isakson on 14 Sep 2014

rkkkkkkk.fig

y''=xy, y(0)=0, y'(0)=1, (Airy equation) step size h=0.05 and interval is [-10,2], this is system of ode, I can already solve runge kutta method but also i need airy solution I can try but not make correct graph please help me...

    function [x, u]= RK4_sys100(f, tspan, u0, n)
    a=tspan(1); 
    b=tspan(2);
    h=(b-a)/n;disp(h);
    x=(a:h:b)';
    u(1,:)=u0;
    for i=1:n
        k1=h*feval(f, x(i),     u(i,:)     )';
        k2=h*feval(f, x(i)+h/2, u(i,:)+k1/2)';
        k3=h*feval(f, x(i)+h/2, u(i,:)+k2/2)';
        k4=h*feval(f, x(i)+h,   u(i,:)+k3)';
        u(i+1, :)=u(i, :)+ (k1/6 +k2/3 +k3/3 +k4/6);
    end
    %x=x;
    %u=u;

this is my system of RK4 coad

    clf; 
    clear u;clear x;
    %forward=======================
    f=inline('[u(2); x*u(1)]','x','u');
    [x, u]=RK4_sys10(f, [0,2],[0,1], 40);
    plot(x,u(:,1),'k') % i can take u(2)
    hold on;grid on;
    %backward===========================
    clear u;clear x;
    f=inline('[u(2); x*u(1)]','x','u');
    [x, u]=RK4_sys10(f, [0,-10],[0,1], 200);
    plot(x,u(:,1),'k') % i can take u(2)
    %hold on;grid on;
    %plot(x,u(:,2))
    %disp('   x    u(1)   u(2)')
    %disp([x u])
    %axis([-10 4, -4 5]);   %points of x and y axis
    x=-10:0.05:2;
    %plot(x,airy(2,x),'r')not this one=================   
    a1=airy(0,x);
    a2=airy(2,x);
    p=3^(2/3)*gamma(2/3);   q=3^(1/3)*gamma(1/3);
    r=3^(1/6)*gamma(2/3); s=3^(-1/6)*gamma(1/3);
    z=[r/p*s+1/q]; y2=((r/p)*a2-a1)/z;
    plot( x,y2,'-b'); %if i can not write x,ai,'-k' then plot line is one
    axis([-10 4, -4 5]);   %points of x and y axis