Solving a non-linear second order differential equation on Scilab?

Question

I need to solve the following

-cos(y)y''+sin(y)y'^2+sin(y)=0, y'(0)=y'(1)=0, such that y=y(t)

I find it hard to solve because of the term y'^2 and also the boundary conditions.

Answer 1

Here is the Scilab code for your bvp

-cos(y)y''+sin(y)y'^2+sin(y)=0, y'(0)=y'(1)=0, y(0)=0, y(1)=1.5

but with different boundary condition not giving the trivial solution. You have first to write y'' as a function of y and y'. The function fsub computes y'' as a function of u=[y,y']

function ysecond=fsub(x,u)
    y=u(1);
    yprime=u(2);
    ysecond = sin(y)/cos(y)*(1+yprime^2);
end

function g=gsub(i, u),
    y=u(1);    
    select i
      case 1 then  // x=zeta(1)=0
        g = y // y(0)=0
      case 2 then // x=zeta(2)=1
        g = y-1.5 // y(1)=1.5
    end
end

N=1;
m=2;
x_low=0
x_up=1;
xpoints=linspace(0,1,100);
zeta=[0,1];


u = bvodeS(xpoints,m,N,x_low,x_up,fsub,gsub,zeta)
plot(xpoints,u(1,:))