594 CHAPTER 30. LAPLACE TRANSFORM METHODS

Note how there is no initial condition. We just look for all solutions to the abovedifferential equation. The following theorem describes all of these solutions.

Theorem 30.3.2 The general solution to the homogeneous problem x′ = Ax consists of allvectors of the form Φ(t)c where c is a vector in Fn and Φ(t) is the fundamental matrix ofA.

Proof: Let x be a solution to the equation. Then x(0) = c for some c. Consider Φ(t)cand x(t) both solve x′ = Ax the first doing so because

Φ′ (t)c= AΦ(t)c

Thus Φ(t)c and x(t) both solve the same differential equation and have the same initialcondition. Therefore, these are the same and this shows that the set of solutions to x′ = Axconsists of Φ(t)c for c ∈ Fn as claimed. ■

Example 30.3.3 Find the general solution to x′ = Ax where

A =

4 1 2 1−3 0 −2 −1−7 −3 −4 −38 4 6 5

According to the above theory, it suffices to find the fundamental matrix. The inverse

of sI−A is the matrix which has the following columns, beginning at the left and movingtoward the right:

s+2s2−2s+1− 3

s2−2s+17s−13

−s3+4s2−5s+2− 1

3s−5 (4s−7) 6s−10−s3+4s2−5s+2

 ,

1

s2−2s+1s−2

s2−2s+13s−5

−s3+4s2−5s+2− 2s−3

3s−56s−10

−s3+4s2−5s+2

 ,

2

s2−2s+1− 2

s2−2s+1

− s2−8s+11−s3+4s2−5s+2− 6s−10−s3+4s2−5s+2

 ,

1

s2−2s+1− 1

s2−2s+13s−5

−s3+4s2−5s+2

− s2+s−4−s3+4s2−5s+2

This was done by a computer algebra system. Now take inverse Laplace transforms of thisto get the fundamental matrix Φ(t) =

et (3t +1) tet 2tet tet

−3tet −et (t−1) −2tet −tet

et − e2t −6tet et − e2t −2tet 2et − e2t −4tet et − e2t −2tet

2e2t −2et +6tet 2e2t −2et +2tet 2e2t −2et +4tet 2e2t − et +2tet

therefore, the general solution is of the form Φ(t)c where c ∈ Fn. In other words, it is theset of linear combinations of the columns of Φ(t). Since Φ(t)−1 = Φ(−t) , the columns