Running ode45 from problem statement

ok so my problem statement is "Use the MATLAB built-in function ode45 to perform the same calculations and generate the plots (study Example 10-11)." I have attatched the picture of the problem statement as well and I was able to run the " Sys2ODEsRK4" No problem but Im having trouble with the ode45 :

This is the Sys2ODEsRK4 code

function [t,y,ydot] = Sys2ODEsRK4(ODE1,ODE2,a,b,h,x1,y1)

t(1) = a; y(1) = .025; ydot(1) = 0;
n= (b-a)/h;
for i=1:n
    t(i+1) = t(i) + h;
    tm = t(i) + h/2;
    Kx1 = ODE1(t(i),y(i),ydot(i));
    Ky1 = ODE2(t(i),y(i),ydot(i));
    Kx2 = ODE1(tm,y(i)+ Kx1*h/2,ydot(i)+Ky1*h/2);
    Ky2 = ODE2(tm,y(i)+ Kx1*h/2,ydot(i)+Ky1*h/2);
    Kx3 = ODE1(tm,y(i)+ Kx2*h/2,ydot(i)+Ky2*h/2);
    Ky3 = ODE2(tm,y(i)+ Kx2*h/2,ydot(i)+Ky2*h/2);
    Kx4 = ODE1(t(i + 1),y(i)+ Kx3*h,ydot(i)+Ky3*h);
    Ky4 = ODE2(t(i + 1),y(i)+ Kx3*h,ydot(i)+Ky3*h);
    y(i+1) = y(i)+(Kx1 + 2*Kx2 + 2*Kx3 + Kx4)*h/6;
    ydot(i+1) = ydot(i)+(Ky1 + 2*Ky2 + 2*Ky3 + Ky4)*h/6;
end

Now for my script file part two I got the correct graphs

clc; clear all 
a=0;b=10;h=0.02;

[t,y,ydot]=Sys2ODEsRK4(@dydt,@dydotdt,a,b,h,0.025,0);

figure(1)
plot(t,y,'LineWidth',2)
xlabel('Time(s)')
ylabel('Displacement (m)')
title('Displacement over time')
grid on

figure(2)
plot(t,ydot,'LineWidth',2)
xlabel('Time(s)')
ylabel('Velocity (m/s)')
title('Velocity vs Time')
grid on 

function dydx=dydt(t,y,ydot)
    dydx=ydot;
end

function dydx=dydotdt(t,y,ydot)
L=0.1;g=9.80;
dydx=(-2*g/L)*y;
end 

Now when I do my ode45 part this is where im getting my error in the line 6 of this code:

%% Solution using matlab ode45() function

timespan = [0 10];
initCond = 0;
[t,y,ydot] = ode45(@dydt,@dydotdt,timespan,initCond); 
%% plotting results
figure
plot(t,y(:,1));
xlabel('time(s)');
ylabel('Displacement (m)');
title('ode45: Displacement vs time');
figure
plot(t,y(:,2));
xlabel('time(s)');
ylabel('Velocity (m/s)');
title('ode45: Velocity vs time');
 
function dydx=dydt(t,y,ydot)
    dydx=ydot;
end

function dydx=dydotdt(t,y,ydot)
L=0.1;g=9.80;
dydx=(-2/L*g)*y;
end 

Now this is the example in the textbook for ode45 if this helps but Im lost on what im doing wrong:

function dNdt = PopRate(t,N)
bG = 1.1; bL = 0.00025; dG = 0.005;dL = 0.7;
f1 = bL*N(1)*N(2)-dL*N(1);
f2 = bG*N(2) - dG*N(2)*N(1);
dNdt = [f1;f2];

tspan = [0 25];
Nini = [ 500 3000];
[Time Pop] = ode45(@PopRate, tspan, Nini);
plot(Time,Pop(:,1),'-',Time,Pop(:,2),'--')
xlabel('Time (yr)')
ylabel('Population')
legend('Lions','Gazelles')

Answer 1

[Peter O]

Hi Daniel,

For your problem, you were very close and had the solutions to each equation worked out correctly. The function signature for ode45 does not spit back the differential. This is a situation where you can take advantage of formatting the differential equations as a vector and have MATLAB solve them simultaneously. Essentially, you have a two state system, where one state is the derivative of the other, but no state needs to be integrated more than once at any timestep. That is:

dx = f(x), where x is a 2-element vector having x(1) equal to y and x(2) as y_dot. Then the solution becomes:

y = x(1);

ydot = x(2);

dx(1) = ydot;

dx(2) = -2gy/L

This is a very powerful trick that is used routinely in solving complex systems.

%% Solution using matlab ode45() function
timespan = [0 10];
initCond = [0.025, 0]; % y, ydot, grouped together
[t,y] = ode45(@dydt,timespan,initCond); 
%% plotting results
figure
plot(t,y(:,1));
xlabel('time(s)');
ylabel('Displacement (m)');
title('ode45: Displacement vs time');
figure
plot(t,y(:,2));
xlabel('time(s)');
ylabel('Velocity (m/s)');
title('ode45: Velocity vs time');
 
function dx=dydt(t,x)
    % Compute both derivatives. The states (y and ydot) come in as the x vector, of size 2x1 for time t
    L=0.1;
    g=9.80;
    y = x(1);
    ydot = x(2);
    
    dx = zeros(2,1);
    dx(1) = ydot;
    dx(2) =(-2/L*g)*y;
    
end 

[Yo mama]

First of all thank you for taking the time to look at this but when I run this code it still says that there is an error in:

[t,y] = ode45(@dydt,timespan,initCond);

Error in ode45 (line 4)

[t,y] = ode45(@dydt,timespan,initCond);

[Yo mama]

I also get this error

"Error in ode45>dydt (line 21)

y = x(1);

[Peter O]

It seems to run OK here from a single file using MATLAB's ability to nest a function within a script. Is it potentially calling an old version of dydt? The variable x is undefined in your original code. Type

which dydt

and make sure it's executing the file you expect.

[Yo mama]

Oops it was trying to access a previous file! Thanks foo! Thank you for taking the time out of your day to help me!!!