Integral in odefun bvp4c

I am working on the solution of the following 4th order ode:

y'''' + y'' * integral( y' ^ 2 ) = 10

b.c.: y(0) =0; y'(1)=0; y''(1)=0; y'''(1)=0;

How can I write my last equation in @myode?

Thanks in advance for your help.

% ---------------------------------------------------------------------

function fun

clc;

clear;

close all;

%y(4)+y(2)*integral(y(1)^2,0,L)=10

solinit = bvpinit(linspace(0,1,10),[0;0;0;0]);

sol=bvp4c(@myode,@mybc,solinit);

figure;

plot(sol.x,sol.y);

% ---------------------------------------------------------------------

function dydx = myode(x,y)

dydx = [y(2)

y(3)

y(4)

% 10-y(3)*integral(y(2)^2,0,1)

];

% ------------------------------------------------------------

function bc = mybc(ya,yb)

bc = [ya(1)

ya(2)

yb(2)

yb(3)];

Answer 1

[Michael Gareis]

I would try preparing your equation first by dividing by y'':

y''''/y'' + integral( y' ^ 2 ) = 10/y''

then derive it (consider quotient rule!), to eliminate the integral:

(y'''''*y'''-y'''*y'''')/y''^2+y'^2 = -10y'''/y''^2

and multiplying by y''^2:

y'''''*y'''-y'''*y'''' + y'^2 = -10y'''

Which is a 5th order equation. Reduced to a 1st order equation you should get this set of 5 equations:

dydx = [y(2);

y(3);

y(4);

y(5),

(-10y(4)-y(2)^2-y(4)*y(5))/y(4)];

You now need to set a 5th BC and extend your initial guess by one column to solve your problem.

[Pierluigi Auriti]

I only have a doubt: when I multiply by y''^2 my result is:

y'''''*y''-y'''*y'''' + y'^2 *y''^2 = -10y''' right?

I really appreciate your help. Thank you!

[Michael Gareis]

Yes sorry, should be y'''''*y''-y'''*y'''' + y'^2*y''^2 = -10y'''

Since y itself doesn' show up in your equation, you can further simplify it by defining for example g=y'. And work with derivatives of g instead, eventually reducing the order by 1.