int(abs(x-2), x);

plot(%, x=-1..5);

restart;

a := miu1(t) = diff(lambda1(t),t) + lambda3(t):

b := miu2(t) = diff(lambda2(t),t) + lambda1(t) - 4*lambda3(t):

c := miu3(t) = diff(lambda3(t),t) + lambda2(t) - 3*lambda3(t):

d := miu(t) = 2*diff(lambda3(t),t$2) - 5*diff(lambda3(t),t) + 7*lambda3(t):

e:=diff(c, t):

S:=subs({diff(lambda1(t),t)=A,diff(lambda2(t),t)=B,diff(lambda3(t),t)=C,

diff(lambda3(t),t,t)=E}, {a,b,c,d,e}):

eliminate(S, {A,B,C,E});

op(%[2])/5;

I also don't understand the Question 2.

There are different ways to solve this problem. For example, it is a sequence of digits after the decimal point in the expansion of the fraction  248590/1369863  to decimal fraction:

evalf[48](248590/1369863);

                         0.181470701814707018147070181470701814707018147070

restart;

ys := [1.8*a+.4000000000*b+103.9*c-.5000000000*e-102.6*g = 0, .5000000000*b+102.6*c+.3*d+.3500000000*e-102.1*g = 0, -3.3*a-1.050000000*b+102.4*c+.7*d+.2500000000*e-105.1*g = 0, -2.5*a+1.450000000*b+102.6*c+3.3*d+1.050000000*e-102.4*g = 0, -1.6*a+1.100000000*b+105.8*c+.3*d+1.750000000*e-105.5*g = 0, -.7*a-.2500000000*b+105.1*c-.2*d+.2000000000*e-105.7*g = 0, -.3*a-.3500000000*b+102.1*c+2.5*d-1.450000000*e-102.6*g = 0, .2*a-.2000000000*b+105.7*c+1.6*d-1.100000000*e-105.8*g = 0]:

B := [a, b, c, d, e, g]:

A:=Matrix([seq([seq(coeff(lhs(ys[i]), B[j]), j=1..nops(B))], i=1..nops(ys))]);

Your system has infinitely many solutions, depending on two parameters  a  and  g .

The decision with checking:

sys:=[1.8*a+.4000000000*b+103.9*c-.5000000000*e-102.6*g = 0, .5000000000*b+102.6*c+.3*d+.3500000000*e-102.1*g = 0, -.3*a-.3500000000*b+102.1*c+2.5*d-1.450000000*e-102.6*g = 0, -2.5*a+1.450000000*b+102.6*c+3.3*d+1.050000000*e-102.4*g = 0, -3.3*a-1.050000000*b+102.4*c+.7*d+.2500000000*e-105.1*g = 0, -.7*a-.2500000000*b+105.1*c-.2*d+.2000000000*e-105.7*g = 0]:

sol:=solve(sys);

eval(sys, sol);

isolve(24*s+33*t=9);

isolve(6*x+10*y+15*z=7);

_Z1, _Z2  -  any integers

restart;

for N while is((1/6)*Pi^2-sum(1/n^2, n = 1 .. N) >= 0.1e-2) do

od;

N; 

                              1000

First - collect, then - exp:

restart;

a0:=I*(m*x+mp*x-omega[m]*t-omega[mp]*t);

a1:=exp(collect(a0, [x,t]));

     # or

a2:=exp(a0);

applyop(collect, 1, a2, [x, t]); 

Or use  a more powerful  algsubs  command:

w:=x+2+sin(x+2);

algsubs(x+2=u, w);

          w := x+2+sin(x+2)
                  u+sin(u)

PP:=proc(S::set)

local It, Sp, s;

if S={} then return {{}} fi;

It:=proc(Sp, s)

{op(Sp), seq(Sp[i] union {s}, i=1..nops(Sp))}

end;

Sp:={{}};

   for s in S do

   Sp:=It(Sp, s)

   od;

Sp;

end;

Example:

PP({a,b,c,d,e});

{{}, {a}, {b}, {c}, {d}, {e}, {a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}, {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a, b, c, d}, {a, b, c, e}, {a, b, d, e}, {a, c, d, e}, {b, c, d, e}, {a, b, c, d, e}}

X:=Array([seq([x, x^2, 8-4*x], x=0..2.9, 0.1)]);

plot([seq(X[i,2],i=1..30)], [seq(X[i,1],i=1..30)]);

             # or

plot([seq(X[i,3],i=1..30)], [seq(X[i,1],i=1..30)]);

See  http://www.physics.arizona.edu/~restrepo/475A/Notes/sourcea-/node23.html

I have reduced the equation by  x^2  removing the trivial root  x=0.

plots[implicitplot](x+omega^2*x^3-omega=0, omega=-2..10, x=-1..1, thickness=2, numpoints=20000);

It is easy to guess the rule: if  n  is even then  F(n)=F(n-1)*3  else  F(n)=F(n-1)+9  so

F:=proc(n)

option remember;

if n=1 then 3 else

((-1)^n+1)/2*3*F(n-1)+(1-(-1)^n)/2*(F(n-1)+9) fi;

end:

seq(F(n), n=1..15);

                3, 9, 18, 54, 63, 189, 198, 594, 603, 1809, 1818, 5454, 5463, 16389, 16398

Enequality  n^2<=2^n  is true for all integer  n>=4 .

Two variants:

restart:

is(n^2<=2^n) assuming n::integer, n>=4;

restart:

assume(n::integer, n>=4):

is(n^2<=2^n);

                       true

                       true