> 
# A nonhomogeneous 2nd order ODE with constant coefficients is 
# solved from beginning to end using the Method of Undetermined 
# Coefficients.
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# 
# Art Belmonte
# Thu, 15/Feb/96
# Math 308-509 (Worksheet B)
# Text, Sections 3.6, 4.3: Another problem involving the Method of 
# Undetermined Coefficients.
# 
# The LZ procedure provides a short cut to construct linear DEs with
# the auxillary equation.
# 
> unprotect(LZ):\
LZ:=proc(c)\
local a, d, k, L, n, p, r, s, x, y, Z;\
n:=nops(c) - 1;\
a:=array(0..n, c);\
p:=0; s:=0;\
for k from 0 to n do\
    p:=p + a[k]*r^(n-k);\
    s:=s + a[k]*(D@@(n-k))(y)\
od;\
L:=unapply(s, y);\
Z:=unapply(p, r);\
#d:=convert(L(y)(x)=0, diff);\
[eval(L), eval(Z)]\
end;\
protect(LZ):
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# Ex 1-0 (M-103/2): Solution to 2nd order linear nonhomogenous 
# IVP; setup.
> unassign('y'); setup:=LZ([1, 4, 5]);\
L:=setup[1]; Z:=setup[2];\
homog_deq:=convert(L(y)(x)=0, diff);\
nonhomog_deq:=convert(L(y)(x)=60*exp(-2*x)*sin(x), diff);
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# Ex 1-1: Find general solution, yc, of the corresponding 
# homogeneous problem.
> rt:=solve(Z(r)=0, r);\
yc:=unapply(c1*exp(-2*x)*cos(x)+c2*exp(-2*x)*sin(x), x);\
check:=simplify(subs(y(x)=yc(x), homog_deq));
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# Ex 1-2: Ensure that (the terms of) g(x), the RHS in the 
# nonhomogeneous DE, is of the form in Table 3.6.1 (T-159) or 
# Table 4.3.1 (T-204). Otherwise, use the method of variation of 
# parameters. It is in this case, the method of undetermined 
# coefficients is the ticket!
> rhs(nonhomog_deq);
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# Ex 1-3: Set up the subproblems, one for each term in g(x).
> unassign('y');\
nh1:=nonhomog_deq;
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# Ex 1-4: For each subproblem, assume the form of the particular 
# solution to the corresponding nonhomogeneous subproblems. 
# Looking at the Notes beneath Table 3.6.1, we see that s = 1 = 
# #times alpha+beta*I = -2+1*I is a root of the characteristic equation 
# Z(r)=0.
> yp1:=unapply(   x* ( a*(exp(-2*x)*cos(x))\
 + b*(exp(-2*x)*sin(x))),   x);
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# Ex 1-5: Find the particular solution to each subproblem. Then sum 
# them to form a particular solution to the full nonhomogeneous 
# problem. (HELLO! -> THIS IS THE NEW IDENTITY 
# CONSTRUCT THAT TOM KIFFE INTRODUCED IN 
# CHAPTER 6 OF _SOLVING ODES WITH MAPLE V_; MUCH 
# SLICKER THAN THE OLD WAY!)
> simplify(subs(y(x)=yp1(x), nh1));\
sol:=solve(identity(", x), {a, b});\
yp1:=unapply(subs(sol, yp1(x)), x);
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# Ex 1-6: Finally, obtain the general solution to the full 
# nonhomogeneous problem by adding the general solution to the 
# homogeneous problem and the particular solution to the full 
# nonhomogeneous problem.
> y:=unapply(yc(x)+yp1(x), x);\
check:=simplify(nonhomog_deq);
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# "Victory at sea..."
>