error "Error in untitled (line 9)"

I'm trying to run the following code, byt I have the error "Error in untitled (line 9) [t,x]=ode45('prob',tspan,x0);" .

Why is that happening? Can anyone help me, please? Thank you!

%Numerical Solutions
%Problem #57
clc
clear
close all
%Numerical Solution
x0=[0;0];
tspan=[0 15];
[t,x]=ode45('prob',tspan,x0);
figure(1)
plot(t,x(:,1));
title('Problem #57');
xlabel('Time, sec.');
ylabel('Displacement, m');
hold on
%Analytical Solution
m=100;
c=20;
k=1000;
F=30;
w=sqrt(k/m);
d=c/(2*w*m);
wd=w*sqrt(1-d^2);
to=1;
phi=atan(d/sqrt(1-d^2));
%for t<to
t=linspace(0,1,3);
x=0.*t;
plot(t,x,'*');
%for t>=to
t=linspace(1,15);
x=F/k-F/(k*sqrt(1-d^2)).*exp(-d.*w.*(t-to)).*cos(wd.*(t-to)-phi);
plot(t,x,'*');
legend('Numerical', 'Analytical')
%M-file for Prob #50
function dx=prob(t,x)
[rows, cols]=size(x);dx=zeros(rows, cols);
m=100;
c=20;
k=1000;
F=30;
if t<1
dx=0;
else
dx(1)=x(2);
dx(2)=-c/m*x(2) - k/m*x(1) + F/m;
end
end

Answer 1

[Stephen23]

The ODE45 documentation states that its first input argument must be a function handle:

https://www.mathworks.com/help/matlab/ref/ode45.html#bu00_4l_sep_shared-odefun

Therefore you should provide its first input argument as a function handle. That will avoid the error that you show... but then you will get other errors, due to the sizes of the output returned by that function... I fiddled around to avoid those (i.e. you need to check):

%Numerical Solutions
%Problem #57
%Numerical Solution
x0=[0;0];
tspan=[0 15];
[t,x]=ode45(@prob,tspan,x0);
figure(1)
plot(t,x(:,1));
title('Problem #57');
xlabel('Time, sec.');
ylabel('Displacement, m');
hold on
%Analytical Solution
m=100;
c=20;
k=1000;
F=30;
w=sqrt(k/m);
d=c/(2*w*m);
wd=w*sqrt(1-d^2);
to=1;
phi=atan(d/sqrt(1-d^2));
%for t<to
t=linspace(0,1,3);
x=0.*t;
plot(t,x,'*');
%for t>=to
t=linspace(1,15);
x=F/k-F/(k*sqrt(1-d^2)).*exp(-d.*w.*(t-to)).*cos(wd.*(t-to)-phi);
plot(t,x,'*');
legend('Numerical', 'Analytical')

%M-file for Prob #50
function dx=prob(t,x)
m=100;
c=20;
k=1000;
F=30;
dx = [x(2);-c/m*x(2)-k/m*x(1)+F/m];
end

[felipe]

Thank you for your help! It works.

Although, I found some simpler solution.

Define [t,x] = ode45(@(t, x) prob(t, x), tspan, yo);

Will fix it.

[Stephen23]

"Although, I found some simpler solution. "

It is unclear to me how you think that creating an anonymous function like this:

[t,x] = ode45(@(t, x) prob(t, x), tspan, yo);

is a "simpler solution" than the (actually simpler) function handle that I showed in my answer:

[t,x] = ode45(@prob, tspan, x0);

It seems that perhaps we use different meanings of the word "simpler".

[Walter Roberson]

@prob will execute more quickly than @(t,x) prob(t,x)

f1 = @prob

f1 = function_handle with value:
    @prob

f2 = @(t,x) prob(t,x)

f2 = function_handle with value:
    @(t,x)prob(t,x)

functions(f1)

ans = struct with fields:
    function: 'prob'
        type: 'simple'
        file: ''

functions(f2)

ans = struct with fields:
            function: '@(t,x)prob(t,x)'
                type: 'anonymous'
                file: '/tmp/Editor_utsta/LiveEditorEvaluationHelperEeditorId.m'
           workspace: {[1×1 struct]}
    within_file_path: ''

Mathworks literally calls the first version simple .

Each time an anonymous function handle is called, MATLAB has to go through the work of injecting the values in workspace into the workspace of the called function. That is extra overhead that is not required for simple function handles.

[felipe]

Actually, my solution and yours have the same result. I was trying to solve another equation where dx(2) involved some term explicitly in "t" and the @(t,x) prob(t,x) works fine.

[felipe]

Using the followinf code I got the figure bellow.

%Numerical Solutions
%Problem #57
clc
clear
close all
%Numerical Solution
x0=[0;0];
tspan=[0 15];
[t,x]=ode45(@(t,x) prob(t,x),tspan,x0);
figure(1)
plot(t,x(:,1));
title('Problem #57');
xlabel('Time, sec.');
ylabel('Displacement, m');
hold on
%Analytical Solution
m=100;
c=20;
k=1000;
F=30;
w=sqrt(k/m);
d=c/(2*w*m);
wd=w*sqrt(1-d^2);
to=1;
phi=atan(d/sqrt(1-d^2));
%for t<to
t=linspace(0,1,5);
x=0.*t;
plot(t,x,'*');
%for t>=to
t=linspace(1,15);
x=F/k-F/(k*sqrt(1-d^2)).*exp(-d.*w.*(t-to)).*cos(wd.*(t-to)-phi);
plot(t,x,'*');
legend('Numerical', 'Analytical')
%M-file for Prob #50
function dx=prob(t,x)
[rows, cols]=size(x);dx=zeros(rows, cols);
m=100;
c=20;
k=1000;
F=30;
if t<1
dx==0;
else
dx(1)=x(2);
dx(2)=-(c/m)*x(2) - (k/m)*x(1) + F/m;
%dx = [x(2);-c/m*x(2)-k/m*x(1)+F/m];
%end
end
end

[felipe]

It's absolutely clear that I chose the wrong word to use. I was no intend to underestimate no one.

[felipe]

I got the exactly some answer using the following code, with your suggestions.

Thank you again!

%Numerical Solutions
%Problem #57
clc
clear
close all
%Numerical Solution
x0=[0;0];
tspan=[0 15];
[t,x]=ode45(@prob,tspan,x0);
figure(1)
plot(t,x(:,1));
title('Problem #57');
xlabel('Time, sec.');
ylabel('Displacement, m');
hold on
%Analytical Solution
m=100;
c=20;
k=1000;
F=30;
w=sqrt(k/m);
d=c/(2*w*m);
wd=w*sqrt(1-d^2);
to=1;
phi=atan(d/sqrt(1-d^2));
%for t<to
t=linspace(0,1,5);
x=0.*t;
plot(t,x,'*');
%for t>=to
t=linspace(1,15);
x=F/k-F/(k*sqrt(1-d^2)).*exp(-d.*w.*(t-to)).*cos(wd.*(t-to)-phi);
plot(t,x,'*');
legend('Numerical', 'Analytical')
%M-file for Prob #50
function dx=prob(t,x)
[rows, cols]=size(x);dx=zeros(rows, cols);
m=100;
c=20;
k=1000;
F=30;
if t<1
dx==0;
else
%dx(1)=x(2);
%dx(2)=-(c/m)*x(2) - (k/m)*x(1) + F/m;
dx = [x(2);-c/m*x(2)-k/m*x(1)+F/m];
end
end

[Walter Roberson]

The ODE45 documentation states that its first input argument must be a function handle:

For historical backwards compatibility, ode45 permits the first input argument to be a character vector that is the name of the function to use. However, if that is used, then the function named must be accessible from the command line, so effectively the name used must be the name of a .m file (or similar). In particular, when you use a character vector, ode45 cannot look inside your current file to locate a function with the name given.

[Stephen23]

"For historical backwards compatibility..."

That must be quite historical, for more than ten years the 1st input is only documented as a function handle:

https://web.archive.org/web/20120925105249/http://www.mathworks.com/help/matlab/ref/ode45.html