Solve a nonlinear differential system

Hi everyone,

I have problem to plot the next system for the use of an ode45. The two functions are y(t) and x(t) with "t" the variable. The system of equation is the next one:

ay''' - b*x*y' + w*cos(y) = 0

x' - w*L*sin(y)=0

a,b,w and L are constant

If anyone can help thanks a lot for your time.

Answer 1

[Areej Varamban Kallan]

Hi Rahiti,

I understand that you need help in solving nonlinear system of differential equations using 'ode45'. To use 'ode45' the governing equations should be of the form X' = F(X,t), where X and F are vectors. Your system of equations can be recast into this form as follows:

1. Let x1 = x and x2 = y
2. x1' = wLsin(x2)
3. x2' = x3
4. x3' = x4
5. x4' = (bx1x3-wcos(x2))/a

Define these derivatives in a separate function file (func.m) and pass its handle as an argument to ode45.

   function dxdt=func(t,x)
     a=1;
     b=0.2;
     w=1;
     L=0.1;
     dx1dt=w*L*sin(x(2));
     dx2dt=x(3);
     dx3dt=x(4);
     dx4dt=(b*x(1)*x(3)-w*cos(x(2)))/a;
     dxdt=[dx1dt;dx2dt;dx3dt;dx4dt];
   end

A sample execution is shown below

   tspan = [0 5]      % time interval in which equations are solved
   X0    = [1 1 0 0]  % initial values of x1, x2 , x3 and x4 (or x, y, y', y'')
   [t,X]=ode45(@func,tspan, X0)

Refer to the documentation https://www.mathworks.com/help/releases/R2018a/matlab/ref/ode45.html for more details.

[Rahiti Arapari Hargous]

Thanks a lot for you help