13.5. EXERCISES 345

there exists c ∈ Fn such that there is no solution x to the equation c = Φ(t0)x. Bythe existence part of Problem 38 there exists a solution to

x′ = Ax, x(t0) = c

but this cannot be in the form Φ(t)c. Thus for every t, Φ(t)−1 exists. Next supposefor some t0,Φ(t0)

−1 exists. Let z′ = Az and choose c such that

z (t0) = Φ(t0)c

Then both z (t) ,Φ(t)c solve

x′ = Ax, x(t0) = z (t0)

Apply uniqueness to conclude z = Φ(t)c. Finally, consider that Φ(t)c for c ∈ Fn

either is the general solution or it is not the general solution. If it is, then Φ(t)−1

exists for all t. If it is not, then Φ(t)−1 cannot exist for any t from what was justshown.

41. Let Φ′ (t) = AΦ(t) . Then Φ(t) is called a fundamental matrix if Φ(t)−1 exists forall t. Show there exists a unique solution to the equation

x′ = Ax+f , x(t0) = x0 (13.11)

and it is given by the formula

x(t) = Φ(t)Φ(t0)−1x0 +Φ(t)

∫ t

t0Φ(s)−1f (s)ds

Now these few problems have done virtually everything of significance in an en-tire undergraduate differential equations course, illustrating the superiority of linearalgebra. The above formula is called the variation of constants formula.

Hint: Uniquenss is easy. If x1,x2 are two solutions then let u(t) = x1 (t)−x2 (t)and argue u′ = Au, u(t0) = 0. Then use Problem 38. To verify there exists a solu-tion, you could just differentiate the above formula using the fundamental theoremof calculus and verify it works. Another way is to assume the solution in the form

x(t) = Φ(t)c(t)

and find c(t) to make it all work out. This is called the method of variation ofparameters.

42. Show there exists a special Φ such that Φ′ (t) = AΦ(t) , Φ(0) = I, and supposeΦ(t)−1 exists for all t. Show using uniqueness that

Φ(−t) = Φ(t)−1

and that for all t,s ∈ RΦ(t + s) = Φ(t)Φ(s)

Explain why with this special Φ, the solution to 13.11 can be written as

x(t) = Φ(t− t0)x0 +∫ t

t0Φ(t− s)f (s)ds.