2nd order ode by ode45 or other numerical method

Hi every one,

I am trying to solve below differential equation with matlab:

y''+(2/x)*y'=0.1248y

B.C 1 : y'(0)=0

B.C 2: y(1)=1

x interval: [0 1]

solution interval (or y): [0 1]

I used bvp4c and ode45 but every time I faced with error like "solve the collocation equations -- a singular Jacobian encountered"

this is my code:

function SolveBVP()

solinit = bvpinit([0,1],[0 0]);
sol = bvp4c(@deriv,@bcs,solinit);
x=linspace(0,1,100);
y=deval(sol,x);
plot(x,y(1,:),'b-x');
 
function dYdx = deriv(x,Y)
 
dYdx(1) = Y(2); 
dYdx(2) = 0.1248*Y(1)-(2/x)*Y(2);

function res = bcs(ya,yb)
res = [ ya(2) 
        yb(1)-1];

It would be appricated if any one of you can help me.

best regards

Answer 1

[Ameer Hamza]

You ODE have term 2/x, which becomes infinite at x=0. To avoid this, start the limit a bit above 0 (say 1e-20). Also, it is better to pass the initial guess at several points. Try this

x=linspace(1e-20,1,100);
solinit = bvpinit(x,[1 0]);
sol = bvp4c(@deriv,@bcs,solinit);
y=deval(sol,x);
plot(x,y(1,:),'b-x');

Answer 2

[Rasoul Rahimzadeh Bafti]

Dear Ameer,

Thanks for your kind response and consideration. That was very helpful!!!

Sincerely,

Rasoul

[Ameer Hamza]

I am glad to be of help.