Second order ODE with initial conditions

I'm trying to understand how to plot a signal in matlab without success. I saw that the program doesn't work directly with equations of higher orders, it's necessary to turn into first order equations. I tried to use ODE45 but with no sucess too.

The equation is: D2x + 0.1*Dx + x^5 = 6*sin(t), with t ranging from [0 50]

Initial values: x(0)=2 Dx(0)=3

Could someone please help me understand how to make matlab to solve the issue?

Answer 1

[Walter Roberson]

I worked for a while with this in Maple. The only solution I have found so far is a series solution. It starts x(t) = 2 + 3 * t . The pattern of signs for the terms after that is not immediately obvious to me. Each of the subsequent terms involves an increasing constant times t to a power, so the series is divergent and goes to roughly +/- 10^N for t=50 for truncation order N.

Therefore, unless by working it through you can observe the pattern of powers and convert that to a nice divergent transcendental function, I am not sure you are going to be able to come up with an explicit solution.

[Gustavo Santana]

Thx for the reply, but my difficulty is more a practical problem. I'm not able to plot the solution of the ODE mentioned. I'm trying to learn how to enter commands at matlab. If you can help me I would greatly appreciate it.

Answer 2

[Friedrich]

Hi,

Based on this article (chapter 2.3)

http://www.math.tamu.edu/REU/comp/matode.pdf

I would do something like this:

function example( )

x_0 = [2,3];
tspan = [0,50];
[t,y] = ode45(@my_func,tspan,x_0);

%y(1) = x
%y(2) = x'
%x'' = 6sin(t)-0.1y(2)-y(1)^5

hold on
plot(t,y(1))
plot(t,y(2))
plot(t,6*sin(t) - 0.1*y(2) - y(1).^5)
hold off

%transform 2nd order ode to couples system of 1st order systems
%x_1 = x
%x_2 = x'
%==> x_1' = x_2
%    x_2' = 6sin(t)-0.1x_2-x_1^5

      function out = my_func(t,x)
          out = [x(2); 6*sin(t) - 0.1*x(2) - x(1).^5];
      end

end

[Gustavo Santana]

I understand the operation and basically read the entire file, but do not know why I can't plot the solution. In the case of the above code, I get the message: "Error: Function definitions are not permitted in this context."

I understand the theory but I can not do the run matlab plotting. I'm studying about this software so I'm facing some difficulties.

PS: I use Matlab R2010b

[Walter Roberson]

Function definitions *are* permitted in that context, provided you wrote the whole code to example.m and executed that. You cannot paste code with functions to the command window.

Answer 3

[Gustavo Santana]

Really Function definitions are permitted in that context. After save the .m archive worked well, but the result aren't crrect i think. Below I put a link in the resulting plot (Fig1) and I should find (Fig. 2). Why graphics are not the same? Please, how can I get the plot of Fig 2?

http://imageshack.us/f/43/resultsjr.png/

[Walter Roberson]

I made no claim that Friedrich's code was correct. His plot completely disagreed with the analysis I did in Maple. On the other hand, I did indicate that the path I was following was leading to a divergent solution that could not reasonably be truncated at any order -- which is consistent with the description of the response characteristics as given in that link. Unless I am able to find a finite but unstable representation, I do not see how it could be symbolically constructed.