> 
> 
# A nonhomogeneous 2nd order ODE with constant coefficients is 
# solved from beginning to end using the Method of Undetermined 
# Coefficients.
--------------------------------------------------------------------------------
# 
# Art Belmonte
# Tue, 13/Feb/96
# Math 308-509 (Worksheet A)
# Text, Sections 3.6, 4.3: Lecture: 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):\
\
--------------------------------------------------------------------------------
# Ex 1-0 (S3.6, 162/15): Solution to 2nd order linear 
# nonhomogenous IVP; setup.
> unassign('y'); setup:=LZ([1, -2, 1]);\
L:=setup[1]; Z:=setup[2];\
homog_deq:=convert(L(y)(x)=0, diff);\
nonhomog_deq:=convert(L(y)(x)=x*exp(x)+4, diff);\
IC:={y(0)=1, D(y)(0)=1}; IVP:={nonhomog_deq} union IC;
--------------------------------------------------------------------------------
# Ex 1-1: Find general solution, yc, of the corresponding 
# homogeneous problem.
> rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(x) + c2*x*exp(x), x);\
yc:=eval(y);\
check:=simplify(homog_deq);
--------------------------------------------------------------------------------
# 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);
--------------------------------------------------------------------------------
# Ex 1-3: Set up the subproblems, one for each term in g(x).
> unassign('y');\
nh1:=convert(L(y)(x)=x*exp(x), diff);\
nh2:=convert(L(y)(x)=4, diff);
--------------------------------------------------------------------------------
# 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 = 2 = 
# #times alpha=1 is a root of the characteristic equation Z(r)=0.
> yp1:=unapply(x^2*(a*x+b)*exp(x), x);\
yp2:=x->c;
--------------------------------------------------------------------------------
# Ex 1-5: Find the particular solution to each subproblem. Then sum 
# them to form a particular solution to the full nonhomogeneous 
# problem.
> simplify(subs(y(x)=yp1(x), nh1));\
sol:=solve(identity(", x), {a, b});\
yp1:=unapply(subs(sol, yp1(x)), x);\

--------------------------------------------------------------------------------
# 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)+yp(x), x);\
check:=simplify(nonhomog_deq);
--------------------------------------------------------------------------------
# Ex 1-7: Since this was an IVP, we now determine c1 and c2.
> IC; c_sols:=solve(", {c1, c2});\
final_solution; y:=unapply(subs(c_sols, y(x)), x);\
final_check:=IVP;
--------------------------------------------------------------------------------
# "Victory at sea..."
>