Interesting problem!
I was curious enough to do the entire simulation to see what the results are —
syms x(t) x0 Dx0 T Y
Dx = diff(x);
D2x = diff(Dx);
DEqn = D2x + Dx + x == 1;
xs = dsolve(DEqn, x(0)==x0, Dx(0)==Dx0)
xs = subs(xs, {x0,Dx0},{1,1}) % Numerical Initial Conditions
[VF,Sbs] = odeToVectorField(DEqn)
DEfcn = matlabFunction(VF, 'Vars',{T,Y})
tspan = [0 10];
[t,x] = ode89(DEfcn, tspan, [1 1]);
figure
fplot(xs, tspan, 'DisplayName','Analytic')
hold on
plot(t, x(:,1), 'DisplayName','Numeric')
hold off
grid
xlabel('Time')
ylabel('x_1(t)')
legend('Location','best')
They match almost exactly.
.
