a1 = 10.0;
a2 = 10.0;
L  = 10.0;
dx = .05;

t1 = sqrt(L / ((1.0 + a1/a2)*a1));
t2 = a1*t1/a2;
Xm = a1*t1*t1;

X = 0:dx:L;
n = length(X);
T = zeros(1,n);
Y = zeros(1,n);

for i=2:1:n
    if X(i)<=Xm
        a = a1/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c;
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) + a1*dt;
    else
        a = -a2/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) - a2*dt;
    end
end

X1 = X;
Y1 = Y;
leg1 = ['t = ', num2str(t1+t2,'% 10.4f'), 's; a1 = ', num2str(a1,'% 10.4f'), '; a2 = ',num2str(a2,'% 10.4f')]

a1 = 1.0;
a2 = 3.0;
L  = 100.0;
dx = .05;

t1 = sqrt(L / ((1.0 + a1/a2)*a1));
t2 = a1*t1/a2;
Xm = a1*t1*t1;

X = 0:dx:L;
n = length(X);
T = zeros(1,n);
Y = zeros(1,n);

for i=2:1:n
    if X(i)<=Xm
        a = a1/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c;
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) + a1*dt;
    else
        a = -a2/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) - a2*dt;
    end
end

X2 = X;
Y2 = Y;
leg2 = ['t = ', num2str(t1+t2,'% 10.4f'), 's; a1 = ', num2str(a1,'% 10.4f'), '; a2 = ',num2str(a2,'% 10.4f')]

a1 = 4.9;
a2 = 4.9;
L  = 100.0;
dx = .05;

t1 = sqrt(2*L / ((1.0 + a1/a2)*a1));
t2 = a1*t1/a2;
Xm = a1*t1*t1;

X = 0:dx:L;
n = length(X);
T = zeros(1,n);
Y = zeros(1,n);

for i=2:1:n
    if X(i)<=Xm
        a = a1/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c;
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) + a1*dt;
    else
        a = -a2/2.0;
        b = Y(i-1);
        c = -dx;
        D = b*b - 4*a*c
        dt1 = (-b + sqrt(D))/(2.0*a)
        dt2 = (-b - sqrt(D))/(2.0*a)
        if (dt2>0)
            if (dt1 < dt2)
                dt = dt1
            else
                dt = dt2
            end
        else
            dt = dt1;
        end
        T(i) = T(i-1) + dt;
        Y(i) = Y(i-1) - a2*dt;
    end
end

X3 = X;
Y3 = Y;
//leg3 = ['t = ', num2str(t1+t2,'% 10.4f'), 's; a1 = ', num2str(a1,'% 10.4f'), '; a2 = ',num2str(a2,'% 10.4f')]

plot (X1,Y1,X2,Y2,X3,Y3);
//grid on;
//xlabel('x(t)');
//ylabel('y(t)');

//legend(leg1,leg2,leg3);