Please help me solve this differential equation

please i will like to solve this eqution for w=188 rad/s

Please take a2=3 a3=2, b2=5 and b3=2

[Torsten]

Solve for x_double_dot, then substitute x = y1, x_dot = y2 and write your equation as a system of two first-order differential equations:

y1' = y2
y2' = x_double_dot = ...

Finally, use ODE45 to solve.

[John D'Errico]

You tell us what the value of w is, but not all of the other important parameters. They are just as important. Thus what are a2, a3, b2 b3? If you don't have them, then no numerical solution will be possible. My expectation is no analytical solution will exist in any case, so a numerical solution is your only option.

[Armel Kapso]

Please take a2=3 a3=2, b2=5 and b3=2

Answer 1

[Sam Chak]

Hi @Armel Kapso

Some basic code from the ode45 documentation. Probably looks like this, try manupulating the parameters yourself.

tspan  = [0 1];     % simulation time interval
initv  = [1 0];     % initial values
[t, x] = ode45(@odefcn, tspan, initv);
plot(t, x(:, 1), 'linewidth', 1.5)

grid on
xlabel({'$t$'}, 'Interpreter', 'latex')
ylabel({'$x(t)$'}, 'Interpreter', 'latex')
title('System Response')

function dxdt = odefcn(t, x)
    dxdt    = zeros(2, 1);
    a2      = 3; 
    a3      = 2; 
    b2      = 5; 
    b3      = 2;
    omega   = 188;
    num     = 2*a3*omega^3*sin(x(1) - omega*t)*x(2) + 2*a3*omega*sin(x(1) - omega*t)*x(2)^3 - b3*sin(x(1)) - b3*sin(omega*t);
    den     = 2*a2 - 2*a3*omega*(2*x(2) + omega)*cos(x(1) - omega*t);
    dxdt(1) = x(2);
    dxdt(2) = num/den;
end

[Armel Kapso]

please this code is not going when i try to run it with my MATLAB R2018b

please help me

[Sam Chak]

@Armel Kapso, I don't know what happened because I don't have R2018b. Moreover, the ode45() was introduced before R2006a.

I indistinctly guess that you copied the code and directly pasted on the Workspace, and pressed [ENTER].

[Armel Kapso]

@Sam Chak No i didn't it's the fucntion you defined that i copied but i did not see that it was long i mean 'num' and so it was crossing the space allocated for the new function i wanted to add

that's why it was not working

Thanks a Lot Sir @Sam Chak

[Sam Chak]

I'm glad to hear that you have sorted it out, @Armel Kapso.

[Armel Kapso]

please @Sam Chak help plot the phase space i'm having difficulty in doing it

[Sam Chak]

That's the way to plot the phase.

tspan  = [0 1];
initv  = [1 0];
[t, x] = ode45(@odefcn, tspan, initv);
plot(x(:, 1), x(:, 2), 'linewidth', 1.5)

[Armel Kapso]

thank you

[Armel Kapso]

Hi @Sam Chak please the system i'm studying is actually a chotic system. my challenge now is to plot the Lyapunov exponent. i went through some of the contribution you have shared in that same domain and i have tried to emplement this one but it doesn't work. please help me again.

a2 = 3;

a3 = 2;

b2 = 5;

b3 = 2;

omega = 210;

f = @(t, x) [(2*a3*omega^3*sin(x(1) - omega*t)*x(2) +...

2*a3*omega*sin(x(1) - omega*t)*x(2)^3 +...

- b3*sin(x(1)) - b3*sin(omega*t)/(2*a2 - 2*a3*omega*(2*x(2) + omega)*cos(x(1) - omega*t))];

tspan = linspace(0, 88, 8801);

init = [1 1 1];

[t, x] = ode45(f, tspan, init);

plot3(x(:,1), x(:,2), x(:,3)), view(45, 30)

[Torsten]

Corrected.

a2      = 3; 
a3      = 2; 
b2      = 5; 
b3      = 2;
omega   = 210;
f = @(t,x) [x(2);(2*a3*omega^3*sin(x(1)-omega*t)*x(2)+...
            2*a3*omega*sin(x(1)-omega*t)*x(2)^3-...
            b3*sin(x(1))-b3*sin(omega*t))/(2*a2-2*a3*omega*(2*x(2)+omega)*cos(x(1)-omega*t))];
tspan  = linspace(0, 88, 8801);
init   = [1 1];
options = odeset('RelTol',1e-8,'AbsTol',1e-8);
[t, x] = ode15s(f, tspan, init, options);
plot(x(:,1), x(:,2))

Answer 2

[Chunru]

opts = odeset('RelTol', 1e-2, 'AbsTol', 1e-4);
[t, x] = ode45(@diffeqn, [0 1], [0.1; .1], opts);

whos

  Name           Size            Bytes  Class     Attributes

  cmdout         1x33               66  char                
  opts           1x1              3712  struct              
  t           4369x1             34952  double              
  x           4369x2             69904  double              

plot(t, x(:,2))

function dxdt = diffeqn(t, x)
a2=3; a3=2; b2=5; b3=2; w=188;

% Check this out
dxdt(1, 1) = x(2);
dxdt(2, 1) = ( (2*a3*w^3*sin(x(2)-w*t))*x(1) + ...
               (2*a3*w*sin(x(2)-w*t))*x(1).^3   + ...
               (-b2*sin(w*t) - b3*sin(x(2)))  ) ./ ...
               ( 2*a2 -2*a3*w*(2*x(1)+w)*cos(x(2)-w*t) );
end

[Armel Kapso]

Yes actually it's a chaotic system. a small change in the parameter produces significant effects

i intend to plot the motion curve of the system and also its space phase trajectories for different values of the parameters

my main problem is that i don't have strong knowledge in matlab

[Armel Kapso]

i've tested this code on my MATLA R2018b is not going

[Chunru]

Can you show what is the problem in 2018b, such as error message?

[Chunru]

I did a correction in the code above.

[Armel Kapso]

[Chunru]

You need to save the code in a file, e.g., testdiff.m. Then run the program testdiff.m