Ch
Ch.4 Linear Second Order D.E. [pic]
Linear 2-nd order homogeneous D.E. with constant coefficients
[pic]
The solution is [pic] Note [pic]are linear independent.
The Wronskian [pic]for any [pic]
To find the solution, let [pic], and the characteristics equation is
[pic] [pic] giving two roots [pic]and [pic].
(1) [pic], real,
[pic]
(2) [pic], real,
[pic]
(3) [pic], complex, and [pic],
[pic]
For [pic] and [pic] the solution is
[pic]
Where [pic]is the solution of the homogenous equation (i.e. [pic]), and
[pic]is the particular solution due to [pic]
Thus, the general solution is given by
[pic]
Note: If [pic], determine [pic] for [pic],
and [pic]for [pic]. Then [pic]
Methods to find [pic]
1.Undetermined coefficients: (guess [pic] if [pic] is simple functions)
[pic] [pic]
[pic] [pic]
[pic] [pic]
[pic] or [pic] [pic]
[pic] [pic]
Note: If any term in [pic]is one of the solutions in [pic] or [pic],
then multiply [pic]by [pic]. Ex. If [pic], [pic], and
[pic], try [pic]. In addition, if [pic] & [pic] are
double root, then multiply [pic]by [pic].
2.Variation of parameters (no guessing)
Let [pic],
Solve [pic], where [pic] is the coefficient in [pic]
Then compute [pic] and [pic],
[pic]
Cauchy-Euler equations: [pic]
Solution: [pic], [pic]is the solution for [pic]
To find [pic], consider [pic], then
[pic]
Hence [pic] [pic] [pic] (2nr order const. coeff)
[pic], to get [pic], use [pic]and [pic].
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