Mechanical Engineers Handbook 2011 Part 15 doc

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

... t2

26666

3777

(a) Find the general solution

(b) Find the particular solution with the initial condition

The initial condition

(a) Find the general solution

(b) Find the particular solution with the initial condition