vv

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These are answers submitted by vv

restart;

f := (2*y-1)*(4*y+6*x-3)/(y+3*x-1)^2;
F :=z -> 4*z*(2*z+3)/(z+3)^2;

(2*y-1)*(4*y+6*x-3)/(y+3*x-1)^2

 

proc (z) options operator, arrow; 4*z*(2*z+3)/(z+3)^2 end proc

(1)

simplify( F((y-1/2) / (x-1/6)) )  =  f;

(2*y-1)*(4*y+6*x-3)/(y+3*x-1)^2 = (2*y-1)*(4*y+6*x-3)/(y+3*x-1)^2

(2)

 

 

1. Replace e^(...)  with  exp(...)
   
(or define  e:=exp(1)).
2.  The integral cannot be computed symbolically.
If you are satisfied with numerical values, just assign values to your constants C:=..., ...
and then execute

evalf( Int(f1(g), g=0..infinity) );


 

It's simpler to express the recurrence in terms of 
X[n] = Sum(x[k], k=1..n)
instead of x[n]. Notice that x[n] = X[n] - X[n-1].
So,

rsolve({u(1) = X[1], u(n + 1) = u(n) + (X[n + 1]-X[n] - u(n))/(n + 1)}, u(n));

           X[n] / n

restart;
f := (x1-x2)*(x2-x3)*(x3-x4)*(x4-x1): 
G := x1^2+x2^2+x3^2+x4^2-1:
H:=f + x5*G:
X:=x1,x2,x3,x4,x5:
Groebner:-Basis(diff~(H, [X]), plex(X)):
S:=solve(%, explicit):
nops([S]);      #    40
S[-1], simplify(eval(f, S[-1]));

{x1 = 1/4 + sqrt(3)/4, x2 = sqrt(3)/4 - 1/4, x3 = 1/4 - sqrt(3)/4, x4 = -1/4 - sqrt(3)/4, x5 = 1/4},   -1/8

DirectSearch works well, not very fast though. (The absolute minimum is -0.125).

restart;
f := (x1-x2)*(x2-x3)*(x3-x4)*(x4-x1): 
G := x1^2+x2^2+x3^2+x4^2-1:
with(DirectSearch):
GlobalOptima(f, [G=0,x1>=0]);

[-0.125000010409962, [x1 = 0.183013027400331, x2 = 0.683012548155197, x3 = -0.683012850582002, x4 = -0.183012508983874], 4041]

restart;
eq1 := c[2] = Z^2/(2*(m + 2)):
eq2 := Int((m*(c[2] - x^2/(2*(m + 2))))^(1/m), x = 0 .. Z) = alpha:
IntegrationTools:-Change(eval(eq2,eq1), x=t*Z,t):
combine(value(%)) assuming m>0,Z>0:
Zsol := combine(solve(%, Z, explicit)) assuming m>0;

(It can be expressed using powers, without ln and exp).

After a change of variables, the integral reduces to

F := int(sin(cos(t))*cos(t),t);

so, for a purely transcendental function.
AFAIK for such functions the Risch algorithm is completely implemented in Maple.
Maple does not compute F, so F is not elementary.
It remains of course the question whether F could be expressed using some special functions.

simplify should be enugh, but combine is necessary here.
simplify(combine(r)) assuming x<=0;     # 0

simplify(combine(r)) assuming x::real;  #  x + |x|

restart
#Digits:=15:
e:=1+0.1/(u^2+0.01*u+1):
n:=sqrt(e-1+n0^2):
Nd:=4*n0/n:
Ns:=n/n0:
Rd:=(Nd-1)/(Nd+1):
Rs:=(Ns-1)/(Ns+1):
#v:=-32/12*evalf(Int(Int(16*Pi^4*u^3*z^4*(exp(-4*Pi*u*n0*z*(Rd*(n0^2+1/2)-1/2*Rs))),n0=0..1),u=0..infinity)):
A:=-32/12*16*Pi^4*u^3*z^4*(exp(-4*Pi*u*n0*z*(Rd*(n0^2+1/2)-1/2*Rs))):
V:= Z ->  -32/12*evalf(Int(eval(A,z=Z),[n0=0..1,u=0..infinity], epsilon=1e-3)):
plot(V, 1..10, numpoints=10, adaptive=false);

implicitplot implements many algorithms and has many options.
E.g. adding the option signchange = false, the line (asymptote) disappears.

The (sub)groups are represented by generators, without generating all the elements.
Both S_10 and S_11 have two generators, see Generators(Symm(10)) and Generators(Symm(11))
so using simple algorithms it will be easy to compute the cosets.

g := (x, T) -> T*x + x^2:  #just an example
dgdx:=(x,T) -> D[1](g)(x,T);
dgdx(1, 2);
dgdx(x, 1);
g:='g';
dgdx(x, T);
convert(%,diff);


                dgdx := (x, T) -> D[1](g)(x, T)
                               4
                            1 + 2 x
                             g := g
                         D[1](g)(x, T)
                           d         
                          --- g(x, T)
                           dx        
 

It is obvious that the result of odetest cannot be 0, because sol is a truncated series.
(strictly speaking, it's not a `series`, the type being `+`).
But this works as expected:

restart;
Order:=20:
ode:=x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0:
sol:=dsolve(ode,y(x),'series',x=infinity):
odetest(sol,ode):
asympt(%,x); # just simplifies

            

stats is obsolete; use Statistics

restart;

with(Statistics):

f := PDF( Normal(0, 2*4.47), x );

0.3155422726e-1*2^(1/2)*exp(-0.6255974455e-2*x^2)

(1)

plot(f, x=-20..20);

 

int(x*f, x);

-3.566543986*exp(-0.6255974455e-2*x^2)

(2)

int(x*f, x=0..5);

.5163728648

(3)

 

 

 

Download stat.mw

 

 

################################
# Solution "by hand"
# For simplicity, take a=0, b=1;
################################

restart;

de := diff(u(x),x$4) = Heaviside(x - a)*u(x) - Heaviside(x - b)*u(x);

diff(diff(diff(diff(u(x), x), x), x), x) = Heaviside(x-a)*u(x)-Heaviside(x-b)*u(x)

(1)

a:=0;b:=1;

0

 

1

(2)

dsolve(de, parametric); # wrong

u(x) = piecewise(x < 0, (1/6)*_C1*x^3+(1/2)*_C2*x^2+_C3*x+_C4, x < 1, _C1*exp(-x)+_C2*exp(x)+_C3*sin(x)+_C4*cos(x), 1 <= x, (1/6)*_C1*x^3+(1/2)*_C2*x^2+_C3*x+_C4)

(3)

de1:=eval(de) assuming x<0;
de2:=eval(de) assuming 0<x,x<1;
de3:=eval(de) assuming 1<x;

diff(diff(diff(diff(u(x), x), x), x), x) = 0

 

diff(diff(diff(diff(u(x), x), x), x), x) = u(x)

 

diff(diff(diff(diff(u(x), x), x), x), x) = 0

(4)

s1:=rhs(dsolve(de1)):
s2:=rhs(dsolve(de2)):
s3:=rhs(dsolve(de3)):

S:=piecewise(x<0, eval(s1,[_C1=c1,_C2=c2,_C3=c3,_C4=c4]), x<1, eval(s2,[_C1=c5,_C2=c6,_C3=c7,_C4=c8]), eval(s3,[_C1=c9,_C2=c10,_C3=c11,_C4=c12]));

S := piecewise(x < 0, (1/6)*c1*x^3+(1/2)*c2*x^2+c3*x+c4, x < 1, c5*exp(-x)+c6*exp(x)+c7*sin(x)+c8*cos(x), (1/6)*c9*x^3+(1/2)*c10*x^2+c11*x+c12)

(5)

seq(seq(limit(diff(S,[x$k]), x=p,left) - limit(diff(S,[x$k]), x=p,right), k=0..3), p=[a,b]):

solve([%]):

SOL:=simplify(eval(S, %));

SOL := piecewise(x < 0, (1/6)*(-c5+c6-c7)*x^3+(1/6)*(3*c5+3*c6-3*c8)*x^2+(1/6)*(-6*c5+6*c6+6*c7)*x+c5+c6+c8, x < 1, c5*exp(-x)+c6*exp(x)+c7*sin(x)+c8*cos(x), 1 <= x, (1/6)*(-x^3*c7+(3*c7-3*c8)*x^2+(3*c7+6*c8)*x-5*c7+3*c8)*cos(1)+(1/6)*(c8*x^3+(-3*c7-3*c8)*x^2+(6*c7-3*c8)*x+3*c7+5*c8)*sin(1)-(1/6)*c5*(x^3-6*x^2+15*x-16)*exp(-1)+(1/6)*c6*exp(1)*(x^3+3*x+2))

(6)

indets(SOL, name);

{c5, c6, c7, c8, x}

(7)

plot(eval(SOL,[c5=5,c6=6,c7=7,c8=8]), x=-1..2);

 

 


Download byhand-ode.mw

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