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# Several Second Order differential equations are solved and checked.
# An auxiliary procedure LZ is used to facilitate the solutions.
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# 
# Art Belmonte
# Thu, 08/Feb/96
# Lecture, introducing:
# LZ: Set up for 2nd order homogeneous constant coefficient 
# differential equations
# 
> 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 (S3.1, T-120/7)
> unassign('y'); setup:=LZ([1, -9, 9]);\
L:=setup[1]; Z:=setup[2];\
deq:={convert(L(y)(x)=0, diff)};\
rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(rt[1]*x) + c2*exp(rt[2]*x), x);\
check:=simplify(deq);
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# Ex 2 (S3.1, T-120/11)
> unassign('y'); setup:=LZ([6, -5, 1]);\
L:=setup[1]; Z:=setup[2];\
deq:={convert(L(y)(x)=0, diff)};\
IC:={y(0)=4, D(y)(0)=0}; IVP:=deq union IC;\
rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(rt[1]*x) + c2*exp(rt[2]*x), x);\
eqs:=IC; sol:=solve(eqs, {c1, c2});\
y:=unapply(subs(sol, y(x)), x);\
check:=simplify(IVP);
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# Ex 3 (S3.4, T-142/11)
> unassign('y'); setup:=LZ([1, 6, 13]);\
L:=setup[1]; Z:=setup[2];\
deq:={convert(L(y)(x)=0, diff)};\
rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(Re(rt[1])*x)*cos(Im(rt[1])*x)\
     + c2*exp(Re(rt[1])*x)*sin(Im(rt[1])*x), x);\
check:=simplify(deq);
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# Ex 4 (S3.4, T-142/18)
> unassign('y'); setup:=LZ([1, 4, 5]);\
L:=setup[1]; Z:=setup[2];\
deq:={convert(L(y)(x)=0, diff)};\
IC:={y(0)=1, D(y)(0)=0}; IVP:=deq union IC;\
rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(Re(rt[1])*x)*cos(Im(rt[1])*x)\
     + c2*exp(Re(rt[1])*x)*sin(Im(rt[1])*x), x);\
eqs:=IC; sol:=solve(eqs, {c1, c2});\
y:=unapply(subs(sol, y(x)), x);\
check:=simplify(IVP);
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# Ex 5 (S 3.5, 150/11)
> unassign('y'); setup:=LZ([9, -12, 4]);\
L:=setup[1]; Z:=setup[2];\
deq:={convert(L(y)(x)=0, diff)};\
IC:={y(0)=2, D(y)(0)=-1}; IVP:=deq union IC;\
rt:=solve(Z(r)=0, r);\
y:=unapply(c1*exp(rt[1]*x) + c2*x*exp(rt[2]*x), x);\
eqs:=IC; sol:=solve(eqs, {c1, c2});\
y:=unapply(subs(sol, y(x)), x);\
check:=simplify(IVP);
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# 
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