Solve the IVP y′′′=sin(y2)+y′+cos(t), provided with an appropriate set of initial conditions of your choice, and using Matlab's ODE45 function.

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Jack Allen
Jack Allen on 19 Nov 2021
Answered: Alan Stevens on 19 Nov 2021
I decided to unpit the range [0,10] and the step size 0.1. Im a little lost on what exactly im doing wrong with this situation. The code I have written so far is this but its not working yet. Im also newer to matlab as well so a little context would be amazing!
clc
clear all
%IVP y''' = sin(y^2)+y'+cos(t), over [0,10] y0 = 0.1
%y1=y, y2=y', y3=y'', y1'=y'=y2, y2'=y''=y3
%y''' = y3' = sin(y1^2)+y2+cos(t)
drdt = @(t,y) [y(2);sin(y(1)^2)+y(2)+cos(y(1))];
range = [0,10];
y0 = 0.1;
[t,y]=ode45(dydt,range,y0);
after the code runs it produces this error:
Index exceeds the number of array elements (1).
Error in DB11>@(t,y)[y(2);sin(y(1)^2)+y(2)+cos(y(1))] (line 8)
dydt = @(t,y) [y(2);sin(y(1)^2)+y(2)+cos(y(1))];
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 115)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
Error in DB11 (line 11)
[t,y]=ode45(dydt,range,y0);
Thanks so much in advance

Answers (1)

Alan Stevens
Alan Stevens on 19 Nov 2021
Like this? (Note the IC for d^2y/dt^2 must be consistent with the values chosen for y(0) and dy(0)/dt). Change the IC's and time range to your heart's content!
%IVP y''' = sin(y^2)+y'+cos(t), over [0,10] y0 = 0.1
%y1=y, y2=y', y3=y'', y1'=y'=y2, y2'=y''=y3
%y''' = y3' = sin(y1^2)+y2+cos(t)
% I find it easier to think of with v = dy/dt and w = dv/dt = d^2y/dt^2
dydt = @(t,y) [y(2); y(3); sin(y(1)^2)+y(2)+cos(t)]; %%%%%%% cos(t) or cos(y(1))?
range = [0,1];
y0 = 0.1; % dy/dt = v
v0 = -0.2; % dv/dt = w
w0 = sin(y0^2) + v0 +cos(0); % dw/dt = sin(y^2)+v+cos(t)
IC = [y0, v0, w0]; % You need initial conditions for y1, y2 and y3
[t,Y]=ode45(dydt,range,IC);
y = Y(:,1); v = Y(:,2); w = Y(:,3);
plot(t,y),grid
xlabel('t'), ylabel('y')

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