Variation of parameters for ODE system

Example:

Find fundamental set of solutions for homogeneous problem

A = sym([1 2; 4 3]);

syms y(t) [2 1]

syms C1 C2

sol = dsolve(diff(y)==A*y);

ysol = subs(y(t),sol)

ysol = 

Y1 = subs(ysol,{C1,C2},{1,0})

Y1 = 

Y2 = subs(ysol,{C1,C2},{0,1})

Y2 = 

Psi = [Y1,Y2]

Psi = 

Find by solving the linear system

f = [exp(-t);1]

f = 

Let , this is the solution of the linear system :

w = Psi\f % solve linear system Psi w = f

w = 

With the inverse matrix

inv(Psi)

ans = 

we can write the solution as

w = inv(Psi)*f

w = 

Take the antiderivative to get

u = int(w,t)

u = 

We obtain the particular solution

Y = expand(Psi*u)

Y = 

For comparison: Solve the inhomogeneous ODE with dsolve

sol = dsolve(diff(y)==A*y+f);

ysol = expand( subs(y(t),sol) )

ysol = 

We get a particular solution by plugging in :

Y = expand( subs(ysol,{C1,C2},{0,0}) )

Y =