Mechanical Engineers Handbook Episode 15 pot

then the real part of the solution ~U and the imaginary part ~V are suspected tobe solutions of the equations Let X ˆPniˆ1ciXi be the general solution of the system 2.21... Remark 2.4The

then the real part of the solution ~U and the imaginary part ~V are suspected to

be solutions of the equations

Let X ˆPniˆ1ciXi be the general solution of the system (2.21)

The solution of the nonhomogeneous system (2.20) must be of the form

Pn iˆ1c0

i…t†x1i ˆ f1…t†

Pn iˆ1c0

i…t†x2i ˆ f2…t†

Pn iˆ1c0

6ˆ 0

for all t 2 …a; b† In this case, the matrix

37

X …t† ˆ M …t†Mÿ1…t0†X0: …2:27†The system of equations (2.24) may be written in the form

X …t† ˆ M …t†Mÿ1…t0†X0‡

…t

t0M …t†Mÿ1…s†F …s†ds: …2:29†

THEOREM 2.10 Liouville's Formula

Let W…t† be the Wronskian of solutions X1; X2; ; Xn of the homogeneoussystem of equations (2.21) Then

W …t† ˆ W …t0†e

„t t0

Pn jˆ1 ajj…s†ds

for which the functions X1; X2; ; Xn,

dxi1dt

dxi2

dt

dxindt

ˆ et…1 ‡ t2† 6ˆ 0 for all t 2 R:

Equations (2.32), in this case, are