MATH 557 Homework Set #5
Fall 2015
20. Prove the product rule for matrix functions:
(A(t)B(t))0 = A0 (t)B(t) + A(t)B 0 (t)
(The key point is that the order of factors must be maintained.)
21. Let Φ = Φ(t) be a fundamental matrix for x0 = A(t)x on (a, b) (recall this means
that Φ(t) is nonsingular and Φ0 = A(t)Φ for every t ∈ (a, b)). Show that Ψ = Ψ(t) is
also a fundamental matrix if and only if Ψ(t) = Φ(t)C for some nonsingular constant
matrix C. Along the way you should derive the formula (Φ−1 )0 = −Φ−1 Φ0 Φ−1 .
22. Let
0
1
A(t) =
− 1t 1 +
1
t
a) Show that
1 + t et
Φ(t) =
1
et
is a fundamental matrix for x0 = A(t)x.
b) Verify directly that Liouville’s formula for det Φ(t) holds in this case.
2
c) Use the fundamental matrix to find the solution with initial state x(2) =
.
1
d) Note that Φ(0) is singular. Why does this not contradict the fact discussed in
class that a fundamental matrix will be nonsingular at every point?
23. Consider the n’th order scalar ODE
x(n) = g(t, x, x0 , . . . , x(n−1) )
x(τ ) = y1
x0 (τ ) = y2
. . . x(n−1) (τ ) = yn
As usual denote the solution by x(t, τ, y). Under what conditions on g will it be true
has determinant which is constant in t? (Remember that ∂x
satisfies a linear
that ∂x
∂y
∂y
equation.)