#
# A second order DE is solved completely with application to
# a circuit problem.
#
# 
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# 
# Art Belmonte
# Tue, 20/Feb/96
# Math 308-509
# Text, Section 3.9: Forced Vibrations
# 
> 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 2-0 (S3.9, 190/18): Electric circuit
> L:=1; R:=5000; C:=1/4 / 1000000;\
unassign('Q'); setup:=LZ([L, R, 1/C]);\
LDO:=setup[1]; Z:=setup[2];\
homog_deq:=convert(LDO(Q)(t)=0, diff);\
nonhomog_deq:=convert(LDO(Q)(t)=12, diff);\
IC:={Q(0)=0, D(Q)(0)=0}; IVP:={nonhomog_deq} union IC;
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# Ex 2-1: Find general solution, Qc, of the corresponding 
# homogeneous problem.
> rt:=solve(Z(r)=0, r);\
Qc:=unapply(c1*exp(-1000*t) + c2*exp(-4000*t), t);\
Q:=Qc;\
check:=simplify(homog_deq);
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# Ex 2-2: Ensure that (the terms of) g(t), 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 2-3: Set up the subproblems, one for each term in g(t).
> unassign('Q');\
nh1:=convert(LDO(Q)(t)=12, diff);
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# Ex 2-4: For each subproblem, assume the form of the particular 
# solution to the corresponding nonhomogeneous subproblems.
> Qp1:=unapply(a, t);
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# Ex 2-5: Find the particular solution to each subproblem. Then sum 
# them to form a particular solution to the full nonhomogeneous 
# problem.
> simplify(subs(Q(t)=Qp1(t), nh1));\
sol:=solve(identity(", t), {a});\
Qp1:=unapply(subs(sol, Qp1(t)), t);\
Qp:=Qp1;
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# Ex 2-6: 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.
> Q:=unapply(Qc(t)+Qp(t), t);\
check:=simplify(nonhomog_deq);
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# Ex 2-7: Resolve constants using ICs.
> IC; c_sols:=solve(", {c1, c2});\
final_solution; Q:=unapply(subs(c_sols, Q(t)), t);\
final_check:=simplify(IVP);\
#plot(u(t), t=0..0.6);
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# Ex 2-8: Other requests...
> charge_after_one_thousandth_of_a_second:=Q(0.001);\
charge_after_one_hundredth_of_a_second:=Q(0.01);\
general_charge:=Q(t);\
Limit(Q(t), t=infinity); limiting_charge:=value(");
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>