2nd degree ODE with ode45

Hello everybody,

this is a great occasion to explain MATLAB's ODE solving strategies to less experienced users like me.

I'm trying to solve a 2nd degree ODE u'' = (1/t)*u' - 4*(t^2)*u with ode45; on [1,20].

Inspired by the documentation for ode45, I tried:

function dy = rigid(x,y)
function ddy = rigid(x,dy)
dy = @(x) diff(y)
ddy = @(x) (1/x)*dy(x) - 4*(x^2)*y(x)
tspan = [1 20]
y0 = [sin(1)+cos(1); 1*cos(1)-1*sin(1)]
[X,Y] = ode45(@rigid,tspan,y0)

plot(X,Y)

which my MATLAB accepts without error messages, but doesn't produce any plot. Could you give me a hint what could be wrong?

Thank you!

Answer 1

[bym]

Well, pretty close. You need to define the ODE as a system of first order ODE's realizing the function arguments can be a matrix (example without using anonymous functions):

function dudt = rigid(t,x)
dudt = [x(2);x(2)./t-4.*t.^2*x(1)]; % function named rigid.m

in the script you would then use

tspan = [1 20];
y0 = [sin(1)+cos(1); 1*cos(1)-1*sin(1)];
[X,Y] = ode45(@rigid,tspan,y0);
plot(X,Y);

[Nina]

Thank you! It makes, sense, now, at least, I'm getting an error message, but now, I really don't understand what's the problem.

function du = rigid(t,u)

du = zeros(3,1); %a column vector

du(1) = du(3)./4.*(t^2) - (t.^(-3))*du(2); %this is line 5

du(2) = @(t) du(3).*t + du(1)*4.*(t.^3)

du(3) = @(t) du(2)./t - du(1)*(4.*t.^2)

[T,U] = ode45(@rigid,[1 20],[sin(1)+cos(1) cos(1)-sin(1) -3*cos(1)-5*sin(1)]);

plot(T,U)

Matlab response is "Error using Ch16Pr20b (line 5)

Not enough input arguments."

Do you know what could be the missing argument?

[Nina]

Actually, a more simple code would be

function xdot = fun1(t,x);

xdot(2) = (1/t)* x(2) - 4*t.^2* x(1);

xdot(1) = x(2);

xdot=xdot(:);

X0 = [sin(1)+cos(1) cos(1)-sin(1) -3*cos(1)-5*sin(1)];

t0 = 1;

tf = 20;

options = [];

[t,y] = ode45(@fun1, [t0,tf],X0, options);

plot(t,y)

But the error message remains the same: not enough input arguments...

[Sean de Wolski]

I don't know about the error but you're not going to be able to stack two anonymous functions into a vector du. You will get and error on lines 6/7 for trying this.

[Walter Roberson]

Storing anonymous functions in [object] vectors used to be supported (but not in numeric vectors). But it led to ambiguous syntax, so now one is required to use cell arrays to store anonymous functions.