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

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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
  2 Comments
KSSV
KSSV on 18 Dec 2020
So integrate first w.r.t first, it will be reduced into y'' and then use ode45.
Jon
Jon on 18 Dec 2020
Would this work? ode45 only solves first order equations, wouldn't you still have a second order equation

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Answers (1)

Jon
Jon on 18 Dec 2020
Edited: Jon on 18 Dec 2020
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
  8 Comments
Jon
Jon on 18 Dec 2020
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
James Tursa on 18 Dec 2020
Edited: James Tursa on 18 Dec 2020
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

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