# 2280, Edwards-Penney 4/E, Chapter 5, Exercise 5.3-6 # Cayley-Hamilton-Ziebur method for solving the spring-mass system A:=<-6,4|2,-4>^+;Eye:=<1,0|0,1>;Z:=<0,0|0,0>; LinearAlgebra[Eigenvectors](A); LinearAlgebra[Determinant](r^2*Eye-A); solve(%=0,r); # find roots ; # Euler atoms are: cos(wt), sin(wt), cos(2wt), sin(2wt), w=sqrt(2) # Solution: u = l.c. of the Euler atoms with vector coefficients # Let the vector coefficients be d1, d2, d3, d4 # The answer is = times inverse (W(0)^T) # where W=Wronskian of the four atoms. u0:=; u1:=; # arbitrary coefficients c1, c2, c3, c4 ; IC:=; # = IC.W1, W1:=(W(0)^T)^(-1); w:=sqrt(2); W:=VectorCalculus[Wronskian]([cos(w*t),sin(w*t),cos(2*w*t),sin(2*w*t)],t); W0:=subs(t=0,W); W1:=1/W0^+; C:=IC.W1; #map(diff,C,c1),map(diff,C,c2),map(diff,C,c3),map(diff,C,c4); # Compute u for initial conditions c=1, c2=c3=c4=0 C1:=subs(c1=1,c2=0,c3=0,c4=0,C); # Solution U := C1.;