Second Order Diff Eq with ode45 in Matlab - matlab

Question

So I need to solve x''(t) = -x(t)^p with initial conditions x(0)= 0 and v(0) = x'(0) = v_o = 1.
The value of the parameter p is 1.
This is what I have:
function [t, velocity, x] = ode_oscilation(p)
y=[0;0;0];
% transform system to the canonical form
function y = oscilation_equation(x,p)
y=zeros(2,1);
y(1)=y(2);
y(2)=-(x)^p;
% to make matlab happy we need to return a column vector
% so we transpose (note the dot in .')
y=y.';
end
tspan=[0, 30]; % time interval of interest
[t,velocity,x] = ode45(#oscilation_equation, tspan, 1);
t = y(:,1);
xposition=y(:,3);
velocity=y(:,2);
end
and this is the error message I receive:
ode_oscillation(1)
Error using odearguments (line 91)
ODE_OSCILLATION/OSCILATION_EQUATION must return a
column vector.
Error in ode45 (line 114)
[neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0,
odeArgs, odeFcn, ...
Error in ode_oscillation (line 17)
[t,velocity,x] = ode45(#oscilation_equation, tspan,1);

Answer 1

There's a few things going wrong here. First, from help ode45:
ode45 Solve non-stiff differential equations, medium order method.
[TOUT,YOUT] = ode45(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates
the system of differential equations y' = f(t,y) from time T0 to TFINAL
with initial conditions Y0.
Note that ode45 expects a function f(t,y), where size(t) == [1 1] for time and size(y) == [1 N] or [N 1] for solution values. Your oscilation_equation has the order of input arguments inverted, and you input a constant parameter p instead of time t.
Also, the initial conditions Y0 should have the same size as y; so size(y0) == [N 1] or [1 N]. You just have 1, which is clearly causing errors.
Also, your output arguments t, xposition and velocity will be completely ignored and erroneous, since y is not set as output argument from ode45, and most of all, their names do not correspond to ode_oscilation's output arguments. Also, their order of extracting from columns of y is incorrect.
So, in summary, change everything to this:
function [t, v, x] = ode_oscilation(p)
% initial values
y0 = [0 1];
% time interval of interest
tspan =[0 30];
% solve system
[t,y] = ode45(#(t,y) [y(2); -y(1)^p], tspan, y0);
% and return values of interest
x = y(:,1);
v = y(:,2);
end