How to solve and plot second order differential equation using ode45?

Hey there

Im trying to solve and plot the following differential equation using ode45

x''=(-2k*x-2c*x'-r*omega*(cos(omega*t)+(r/L)*cos(2*omega*t))*m+x0*M)/M+m

where

M = 22

m = 0.9

k = 25000

c = 2

omega = 860

L = 0.2

r = 0.07

The starting conditions are :

x0=0

x'0=0

I've tried a bunch of different tutorials, but keep getting different error messages.

Any help is greatly appreciated, thank you!

[Dyuman Joshi]

Have you looked into the documentation of ode45 to see how to formulate the ODE properly?

"I've tried a bunch of different tutorials, but keep getting different error messages."

Can you share the code you tried and the corresponding errors you got?

[Studentguy]

Ye, but im not sure im understanding it correctly

Here is one of my attempts:

x0 = [0 0];

tspan = 0:.1:10;

ode45(@dxdt,tspan,x0)

plot(t,x)

function dotx = dxdt(t,x)

M = 22;

m = 0.9;

k = 25000;

c = 2;

omega = 860;

l = 0.2;

r = 0.07;

x0_speed = 0;

x0_pos = 0;

dx1=x(2);

dx2=(-2*k*x(2)-2*c*dx1-r*omega^2*(cos(omega*t)+(r/l)*cos(2*omega*t))*m+x0_pos*(M-m))/M;

end

And i got this error message:

Output argument "dotx" (and possibly others)

not assigned a value in the execution with

"Assignment2_a_forsog_2>dxdt" function.

Error in odearguments (line 90)

f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode45 (line 106)

odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in Assignment2_a_forsog_2 (line 4)

ode45(@dxdt,tspan,x0)

[Dyuman Joshi]

Is the value of x0 that appears in the ODE the same as the value of x0 that is one of the initial conditions?

Also, you have defined the variable x_speed in the ODE function, but have not used it. Is there any particular use of that variable?

[Studentguy]

Ye, x0 is the same, its just part the equation i arrived at during calculations.

No x_speed isnt used anywhere.

Answer 1

[Fabio Freschi]

In your code there is a mistake in the definition of x(1) and x(2)

I have made a few stylistic changes (parameters outside the function, use of implicit function) and the correction of the equations. In addition, I suggest to let ode45 to choose the timestep and keep the t vector provided as output for the plot

% clear variables, close all
x0 = [0 0];
tspan = [0 10];

% params
M = 22;
m = 0.9;
k = 25000;
c = 2;
omega = 860;
l = 0.2;
r = 0.07;
x0_speed = 0;
x0_pos = 0;

% implicit function
%                        changes here
%                         |        |
%                         V        V
dxdt = @(t,x)[x(2); (-2*k*x(1)-2*c*x(2)-r*omega^2*(cos(omega*t)+(r/l)*cos(2*omega*t))*m+x0_pos*(M-m))/M];

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

figure
plot(t,x)

figure
plot(t,x)
xlim([0 0.5])

[Studentguy]

Thank you so much!

[Sam Chak]

Hi @Studentguy,

Don't forget to accept Prof. @Fabio Freschi's helpful answer.

[Sam Chak]

Hi @Studentguy

I double-check. The original 2nd-order differential equation in your question

is slightly different from the state equation you defined in your code:

( - 2*k*x(1) - 2*c*x(2) - r*omega^2*( cos(omega*t) + (r/l)*cos(2*omega*t) )*m + x0_pos*(M - m) )/M;

Please clarify!

[Fabio Freschi]

This is why I wrote my code according to the orignal equation (as stated in the comment)