10-39. Show the relation between the vector differential equation
d ̄y
dt =A(t) ̄y+
f ̄(t), y ̄(0) = ̄cand the matrix differential equation
dY
dt =A(t)Y, Y (0) = Iis given by
y ̄=Y(t) ̄c+Y(t)∫t0Y−^1 (ξ)f ̄(ξ)dξ10-40. Verify the forward differences in table 10.1.
10-41. Verify the finite integrals in table 10.2.
10-42. Solve the given difference equations.
(a) yn+2 − 5 yn+1 + 6yn= 0
(b) yn+2 − 6 yn+1 + 9yn= 0
(c) yn+2 − 6 yn+1 + 13yn= 0(d) yn+2 + 4yn+1 + 3 yn= 0
(e) yn+2 + 2yn+1 +yn= 0
(f) yn+2 + 2yn+1 + 10 yn= 010-43. Find the finite integrals
(a) ∆ −^1 x^2 (b) ∆ −^11
(3 + 2x)n(c) ∆ −^11
xn10-44.
(a) For A=
[
2 0
1 3], verify the matrix function eAt =
[
e^2 t 0
e^3 t−e^2 t e^3 t](b) For B=
[
2 1
0 1], verify the matrix function eBt =
[
e^2 t e^2 t−et
0 et]10-45.
(a) Verify the matrix differential equation
dX (t)
dt=AX (t) + F(t), X (0) = Chas the matrix solution
X=X(t) = eAtC+eAt∫t0e−Aξ F(ξ)dξ(b) For A=
[
2 0
1 3],F(t) =
[
et
1]and C=
[
1
0]solve the matrix differential equation
dX (t)
dt =AX +F(t), X (0) = C