Simple script that computes a solution of linear ODEs giving wrong result - matlab

Question

this a question that envolves both programming and mathematics. So, I'm trying to write a code that computes the general solution of a system of linear ODEs described by . The mathematical formula it's shown above:
where the greek symbol \PHI that appers in the equation is the expm(A*t)
clear all
A=[-2]; %system matrix
t0=1; %initial time of simulation
tf=2; %final time of simulation
syms t x_0
x0=x_0;
hom=expm(A*t); %hom means "homogeneous solution"
hom_initialcond=hom*x0;%this is the homogeneous solution multiplied by the initial conditon
invhom=inv(hom); %this is the inverse of the greek letter at which, multiplied by the input of the system, composes the integrand of the integral
g=5*cos(2*t); %system input
integrand=invhom*g; %computation of the integrand
integral=int(integrand,t0,t); %computation of the definite integral from t0 to t, as shown by the math formula
partsol=hom*integral; %this is the particular solution
gen_sol=partsol+hom_initialcond %this is the general solution
x_0=1; %this is the initial condition
t=linspace(t0,tf); %vector of time from t0 to tf
y=double(subs(gen_sol)); %here I am evaluating my symbolic expression
plot(t,y)
The problem is that my plot of the ODE's solution it's not looking well, as you can see:
The solution it's wrong because the curve shown in the graph doesnt start at the initial value equals 1. But the shape it's very similar from the plot gave by the MATLAB ODE solver:
However, if I set t0=0 then the plot gave by my code and by MATLAB solver it's exacly equal to each other. So, my code it's fine for t0=0 but with any other values my code goes wrong.

Answer 1

The general solution in terms of fundamental matrices is
or more often seen as
But since the initial time is often taken to be zero, the inverse of the fundamental matrix is often omitted since it is the identity for linear, constant coefficient problems at zero (i.e., expm(zeros(n)) == eye(n)) and the c vector is equivalent to the initial condition vector.
Swapping some of the lines around near your symbolic declaration to this
syms t x_0 c_0
hom = expm(A*t) ;
invhom = inv(hom) ;
invhom_0 = subs(invhom,t,sym(t0)) ;
c_0 = invhom_0 * x_0 ;
hom_initialcond = hom * c_0 ;
should provide the correct solution for non-zero initial time.