vv

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These are answers submitted by vv

Your ode is

ode := diff(y(x), x) = 15*exp(4*y(x)) - 5*y(x)

There is no explicit solution for this ODE; this is why the Qualitative theory of ODEs exists.
You can study the solutions taking y0 = y(0) as a parameter. You will see that the solution always blows up in a finite interval.

Actually in our case it's easy to see that f(y)>0 (=  rhs(ode)) and

int(1/f(y), y=y0 .. y(x)) = x - 0

so, the existence domain for x>0 is bounded, because LHS is obviously bounded.

P.S. It is a good idea to post the problem, not just in the worksheet; it is short, and the worksheet should be only optional. You will increase the chances to get answers.

Your function has an essential singularity at z=0. Maple (and probably other CAS too) cannot compute the series for such singularities. (The series has the form Sum(a(n)*z^n, n = -infinity .. infinity)).

But you can use Maple to obtain it by multiplying two series (try it first by hand).

This occurs frequently when the symbolic solution is expressed via mathematical functions with branches.

The numerical solution is continuous (in general), and it may coorespond to different branches (i.e. it "jumps" from one branch to another).

``||i   are valid names, so the assignments work.

But note that they are global variables. So, if you have

local a, b__c, t;

then the assignments will produce new global variables, e.g. :-a  will be 1, but a remains free (unassigned).

value(IntegrationTools:-Change(z, s=x-t, t)) assuming x>0;

But do you really need it?

You simply forgot to load the package. Start with:

with(Student[Calculus1]):

 

Selecting the homogeneous solutions is not complicated:

with(PDETools):
PDE := 2*((x1 - x2)^2 + (y1 - y2)^2)*(u1*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), x1) + v1*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), y1) + u2*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), x2) + v2*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), y2)) - ((u1 - v1)^2 + (u2 - v2)^2)*((x1 - x2)*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), u1) + (y1 - y2)*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), v1) - (x1 - x2)*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), u2) - (y1 - y2)*diff(f(x1, y1, x2, y2, u1, v1, u2, v2), v2)) = 0:
n:=2:
P:=rhs(PolynomialSolutions(PDE, degree = n)[]):
C:=[indets(P,suffixed(_C, integer))[]]:
T:=[coeffs(collect(P,C,'distributed'),C)]:
select(u -> degree(u)=n, T); 

[v1^2 + 2*v1*v2 + v2^2, (v1 + v2)*u1 + (v1 + v2)*u2, u1^2 + 2*u1*u2 + u2^2, u1*y2 + u2*y1 - v1*x2 - v2*x1, u1*y1 + u2*y2 - v1*x1 - v2*x2]

But it seems that you want to eliminate also other terms. This will need extra work (probably using Groebner bases).

 

There are essentially two methods.

1. Use DirectSearch.
GlobalOptima(f(x,y), {x=3..4, y=0..1}, maximize);
The approx max is (almost) 0.

2. Use calculus.
Solve the system [diff(f(x,y),x), diff(f(x,y),y)].
Maple finds a unique solution in the domain: x=4, y=0, ( f(4,0)=0 ).
It remains to compute the max on the boundary: f(x,0), f(x,1), f(3,y), f(4,y)
(it's easy).

 

 

Maple has not a package for solving general functional equations.
A few years ago I saw at a conference a group of hugarian mathematicians developping such a package, but I cannot find the references (they should be somewhere on my computer).
Anyway, the equations you mention do not seem to be difficult. The general solution for the first one (for x>0) is:

F(x) = piecewise( x<1, a(x), x^(-r)*a(1/x) )

where a(x) is an arbitrary function for 0 < x <= 1.

@escorpsy 

Your integral diverges. It has a singularity at Zeta=Pi/2. You may check with MultiSeries:-series(phi*W, Zeta = Pi/2)

or compute some values around Pi/2.

P.S. You should not use Zeta as a variable, it is the well known Riemann's function; use zeta instead!

The irreducibility is a very difficult math problem and needs a precise context.
Let's consider the case of polynomials in Z[x,y], which are linear in parameter d (integer too).

restart; # example
randomize():
expand((7+randpoly([x,y]))*(66+randpoly([x,y]))): eval(%,coeffs(%)[5]=d):
p:=%;

p := d*x^7*y^2 - 374*x^9*y - 2510*x^7*y^3 - 1224*x^5*y^5 - 1156*x^9 - 2176*x^5*y^4 + 825*x^4*y^5 + 5100*x^2*y^7 - 2856*x^8 - 1088*x^7*y - 1512*x^6*y^2 + 2550*x^4*y^4 - 3249*x^6*y - 3820*x^4*y^3 + 6300*x^3*y^4 - 2176*x^2*y^5 - 476*x^6 - 544*x^5*y - 422*x^4*y^2 - 288*x^3*y^3 - 4522*x^5 + 1910*x^4*y - 3326*x^3*y^2 + 4488*x^2*y^3 - 2775*x*y^4 + 1200*y^5 + 2244*x^4 - 224*x^3*y + 525*y^4 + 7431*x^3 - 816*x^2*y + 1184*x*y^2 - 512*y^3 - 357*x^2 - 224*y^2 - 2442*x + 1056*y + 462

Here we can try brute force

n:=10000:
for d in {seq(-n..n)} do
  if not irreduc(p) then print('d'=d); break fi od:

                        d = -964

factor(p);

      -(11*x^4*y + 68*x^2*y^3 + 34*x^4 + 84*x^3 - 37*x + 16*y + 7)*(34*x^5 + 18*x^3*y^2 + 32*x^3*y - 75*y^4 + 51*x^2 + 32*y^2 - 66)

You should be pleased about how  '*' and '/' work for inequalities, because this way you will avoid incorrect results.
E.g. try to multiply   -2 < -1   and   1 < 3.


 

 

restart;
with(GraphTheory):
with(SpecialGraphs):
K4 := CompleteGraph(4):
IsPlanar(K4, 'F'):   # true

G:=K4:
n:=max(Vertices(G)):
V1:=[Vertices(G)[],n+1]:
E1:=Edges(G) union {seq({i,n+1},i=1..n)}:
G1:=Graph(V1,E1):
IsPlanar(G1, 'F1'); # iff G is Outerplanar

        false

I had a similar problem with SMTLIB:-Satisfy in Maple 2018 (on Windows 7).

Installing the Visual C++ Redistributable for Visual Studio 2012 Update 4

https://www.microsoft.com/en-us/download/details.aspx?id=30679#

solved the problem.

 

Maple does not miss solutions here. But solve uses the (documented) convention that the arguments of abs are assumed to be real.

Note that the equation has infinitely many complex solutions; they can be found in terms of RootOfs (or even explicitely, but with huge expressions).

However, in this case Maple assumes that the unknown itself is real, not only the argument of abs. (There are 4 complex solutions having the argument of abs equal to -1).

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