Solving coupled second and first order ODEs with ode45

I am looking to numerically solve these coupled ODEs

where both u and η are functions of a variable x only, with , where

are the boundary conditions (with some pre-defined values for α, γ, d, , and ).

I have found multiple different examples of previously asked questions with second-order coupled ODEs similar to mine, but none of them really help me understand the ode45 inputs I am supposed to use for my particular problem...

Answer 1

[darova]

You should use bvp4c

Function for equations

function dy = myode(t,y)
    u = y(1);
    eta = y(2);
    deta= y(3);
    dy(1,1) = (deta-alpha*u^(2+gamma))/u;        % du/dt
    dy(2,1) = deta;                              % deta/dt
    dy(3,1) = u0/u-exp(-eta);                    % d2eta/dt2
end

Function for boundary conditions

function res = bfun(y0,yd)
    res = [y0(1)-u0      % u(0) == u0
            y0(2)        % eta(0) == 0
            y0(3)        % deta(0) == 0
            yd(2)-eta_w];% eta(d) == eta_w
end

p.s. I like the way you organized the question