Backward Eurler and Forward Eurler on matrices in matlab

I'm trying to to solve the sysmtem of differential equations where I'm given initial condition x(0) = X0 and matrix A and use forward Eurler and Backward Eurler to find X.

So I tried writing the code as follows

load ODE_Data %load the matrix A and initial condition X0
dT = 6/500; %sec
t_be = 0:dT:6;
Xbe = zeros(5,length(t_be));
Xbe(:,1) = X0;
for t=2:length(t_be)
    
    Xbe(:,t) = ((1/dT)-A)\(Xbe(:,t-1)/dT);
end

for Backward Eurler using the equation

and

load ODE_Data %load the matrix A and initial condition X0

 %Name your time points as t_fe  and  Name your solution vector as Xfe.
dT = 6/500; %sec
t_fe = 0:dT:6;
Xfe = zeros(5,length(t_fe));
Xfe(:,1) = X0;
for t=2:length(t_fe)
    b =Xfe(:,t-1);
    c=dT*(A+(1/dT));
    Xfe(:,t) = dT*(A+(1/dT))*Xfe(:,t-1);
end

for Forward Eurler using the equation

but the anwsers for both quickly goes to infinity and I'm really lost on why that is.

Answer 1

[Michael Hubatka]

First, make sure that

eig(A) < 0

otherwise, the solution should in fact go to infinity.

The real problem in your code is the calculation of

1/dT - A
A + 1/dT

This will modify all entries in A, the correct code is

eye(size(A))./dT - A
A + eye(size(A))./dT

which will modify only the diagonal.