MATLAB: Solving System of two Differential Equations with one initial and one end condition

initial and end conditionMATLABsystem of two differential equations

This is a code for the heat transfer in a heat exchanger. It solves the system of the two ODEs T_h_dx = … and T_c_dx = … for two initial conditions T_hot_in and T_cold_in.
How can this be solved, if T_cold_in should be the temperature at the end of the heat ecxhanger (x = length / end condition) and T_hot_in still the temperature at x = 0 (initial condition)?
Thank you!
% Functions for heat transfer to be solved by ode
    function T_calc = heat_transfer(~,T)
        T_h = T(1,1);
        T_c = T(2,1);
        T_h_dx = -1 / ((m_hot_plate  / 2) * c_water * R_width) * (T_h - T_c);         
        T_c_dx =  1 / ((m_cold_plate / 2) * c_water * R_width) * (T_h - T_c);      
        T_calc = [T_h_dx ; T_c_dx]; 
    end 
%__________________________________________________________________________                                                                                            
% solve ODE System
    [X, T_tot] = ode23(@(x,T) heat_transfer(x,T), [0 length],[T_hot_in T_cold_in]);

Best Answer

Symbolic appears to work:
syms Th(x) Tc(x)
m_hp = 20;
m_cp = 5;
cw = 1;
Rw = 1;
eq(1) = diff(Th,x) == -1 / ((m_hp / 2) * cw * Rw) * (Th - Tc);
eq(2) = diff(Tc,x) ==  1 / ((m_cp / 2) * cw * Rw) * (Th - Tc);
conds = [Tc(10) == 20, Th(0) == 50];
sol = dsolve(eq,conds);
fplot(sol.Tc,[0 10])
hold on
fplot(sol.Th,[0 10])
hold off
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