New to matlab and not sure how to reduce to first order

Solve the third-order ODE function

y′′′− 2 * y′′−(y′)^2 = 1

With tspan [0 5], y(0) = y’(0) = 0, y’’ = 1.

You need to reduce the given third-order ODE equation to standard 1st-order ODE equations before solving

[KSSV]

So integrate first w.r.t first, it will be reduced into y'' and then use ode45.

[Jon]

Would this work? ode45 only solves first order equations, wouldn't you still have a second order equation

Answer 1

[Jon]

Define

y1 = y

y2 = y'

y3 = y''

% and then make a function f, for example put the code below into an m-file called f.m

function dydt = f(t,y)
dydt(1) = y(2); % y1' = y2
dydt(2) = y(3); % y'' = y3
dydt(3) = 1 + 2*y(3) + y(2)^2;

then use one of the ode solvers e.g ode45 passing it the function f described above

Please see https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html especially the section on solving higher order equation

[James Tursa]

Should ensure that dydt is a column vector also. So either allocate it up front with zeros(3,1) or put dydt=dydt(:) at the end.

[Samantha]

so something like...

function dydt = funcQ1

(t,y)dydt = zeros(3,1);

dydt(1) = y(2);

dydt(2) = y(3);dydt(3) = ...

[James Tursa]

Well, assuming that (t,y) was supposed to be on the first function line, yes.

[Samantha]

[Samantha]

not sure what im doing wrong

[Jon]

1) Rename the file you have called f.m to odef.m (the name of the file and name of the function should match)

2) delete the following lines from the odef.m file

y1 = y
y2 = y'
.
.
.
ode45(odef,...
plot(y,t)

3) make another script, or from the command line to call ode45, so something like

[t,y] = ode45(@odef,[0,5],[0;0;1])
plot(t,y)

note the @ in front of the function name, as you are passing a handle to the function.

[Jon]

You can get documentation for ode45 by typing doc ode45 on the command line.

Once you get this running, if you have not already done so I would highly recommend taking the time to complete the MATLAB on ramp training to help you get more familiar with using MATLAB in general https://www.mathworks.com/learn/tutorials/matlab-onramp.html

[James Tursa]

The ode45( ) call should not be inside your derivative function. That's backwards. The ode45( ) function is calling your derivative function, not the other way around. So you should have some code that defines initial conditions and calls ode45( ) like this:

tspan = [0 5]; % time span
y0 = [0 0 1]; % initial values of y, y', and y''
ode45(@odef,tspan,y0);
plot(t,y);

and then a separate code (e.g., in a file called odef.m, or maybe below the code listed above) that contains your derivative code:

function dydt = odef(t,y)
dydt = zeros(3,1); % the values y', y'', and y'''
dydt(1) = y(2); % y' = y2
dydt(2) = y(3); % y'' = y3
dydt(3) = 1 + 2*y(3) + y(2)^2; % y''' = 1 + 2y'' + (y')^2
end