How does matlab solve a pair of coupled differential equations?

I'm trying to solve a pair of coupled differential equations following the Matlab article in:

https://uk.mathworks.com/help/symbolic/solve-a-system-of-differential-equations.html

When trying to use this example to solve my own equations the code runs but the diff function gives a different function to what I was expecting. I was expecting to get an equation of the form N(t) = N_{0}exp(-t/T). Instead I get N(t) = A - A*exp(-t/T), where A is constant defined by the boundary conditions. This is giving me the wrong time dependence of N(t).

Why am I getting the incorrect solution? Is there something wrong with my code or is there a problem with one of the Matlab functions?

Thanks

Simplified code below.

% Define lifetime variables
T_th = 10e-11;

% set up problems, represent u and v with symbolic functions
syms N_ex(t) N_cb(t) 

% set up the differential equations. 

A = [ -1/T_th 0 ;
    1/T_th 0 ];

B = [N_ex ; N_cb];

ordinary_diff_eqns = diff(B) == A*B

% apply the initial conditions
Initial_conds = B(0) == [1000 ; 0]

% solve the differential equations
[N_exSol(t), N_cbSol(t)] = dsolve(ordinary_diff_eqns, Initial_conds)

%Plot the solutions to see the time dependence.
clf
hold on
fplot(N_exSol)
xlim([0,10e-9])
ylim([0,1000])
xlabel('Time')
ylabel('Population')
fplot(N_cbSol)

Answer 1

[Steven Lord]

From the documentation:

"When solving for multiple functions, dsolve returns a structure by default. Alternatively, you can directly assign solutions to functions or variables by specifying the outputs explicitly as a vector. dsolve sorts outputs in alphabetical order using symvar."

dsolve returns the correct solutions, just not in the order you expect. Call dsolve with one output and extract the appropriate fields of the struct array to view/use the solutions.

[William Royle]

Awesome! Thanks for your help Steven

Answer 2

[Torsten]

Correct solution is

N_ex(t) = 1000*exp(-t/T_th)
N_cb(t) = 1000*(1-exp(-t/T_th))

Best wishes

Torsten.

[William Royle]

But Matlab gives me

N_ex(t) = 1000*(1-exp(-t/T_th))

and

N_cb(t) = 1000*exp(-t/T_th)

Thanks