如何使用 Octave 求解二阶微分方程？

Question

I am completely new to Octave and I am trying to solve the equations below for my project. I was unable to find any straightforward guide to solving the differential equations. Could someone demonstrate how to solve it? Your help is much appreciated!

I am trying to plot a graph of h over time for the equations with a given initial h, miu, r, theta, g, L, and derivatives of h and theta wrt t. Is this possible? And if so, how?

The two equations mentioned

I tried to type the equations into Octave with the given conditions, but there seems to be an error that I am unable to identify.

I am also extremely new to coding, but I need this to complete my project

Answer 1

Whatever is the numerical software you will use, Octave or another one (that does not formal calculation), the first step is to transform your system or N coupled Ordinary Differential Equations (ODEs) of order p into a system of p*N coupled ODEs of order 1 .

This is always done by setting intermediate derivatives as new variables. For your system, then

will become

Then, with Octave, and as explained in doc lsode you define a function say Xdot = dsys(X) , that codes this system. X is the vector [h, theta, H, J] , and Xdot is the returned vector of their respective derivatives, as defined by the right hand expressions of the system of first order ODEs.

So, the 2 first elements of Xdot will be trivial, just Xdot=[X(3) X(4)...] .

Of course, dsys() must also use the parameters M, g, m, µ, L , and r . As far as i understand, they can't be passed as additional arguments to dsys() . So you must defined them before calling lsode .

For initial states, you must define the vector X0=[h0, theta0, H0, J0] of known initial values.

The vector of increasing times >= 0 to which you want to compute and get the values of X must then be defined. For instance, t = 0:100 . 0 must be the first element of t .

Finally, call Xt = lsode(@dsys, X0, t) . After that you should get

• Xt(:,1) are the values of h(t)
• Xt(:,2) are the values of theta(t)
• Xt(:,3) are the values of H(t)=(dh/dt)(t)
• Xt(:,4) are the values of J(t)=(dtheta/dt)(t)