The standard way to solve systems of equations numerically is using fsolve.  For example:

fsolve({x^2-y-3, x*y+2*y^2-4 = 0}, {x = 0.5 .. 2.5, y = 0 .. 3});

#{x = 2.000000000, y = 1.000000000}

That's what the fsolve command will do with it.

infolevel[fsolve]:=6:
fsolve({x^2-y-3, x*y+2*y^2-4 = 0}, {x = 0.5 .. 2.5, y = 0 .. 3});


#fsolve: trying multivariate Newton iteration
#fsolve:
#guess vector Vector(2, [2.1564724756843477,.13446305023563622])
#fsolve: norm of errors: 5.1897839975614897
#fsolve: new norm: 3.6816464474611895
#fsolve: iter = 1 |incr| = 1.4002 new values x = 2.1216 y = 1.4998
#fsolve: new norm: .43227630491510698
#fsolve: iter = 2 |incr| = .53690 new values x = 2.0189 y = 1.0655
#fsolve: new norm: 0.9718475514730430e-2
#fsolve: iter = 3 |incr| = 0.82477e-1 new values x = 2.0005 y = 1.0015
#fsolve: new norm: 0.5297608520287e-5
#fsolve: iter = 4 |incr| = 0.19416e-2 new values x = 2.0000 y = 1.0000
#fsolve: new norm: 0.1568000e-11
#fsolve: iter = 5 |incr| = 0.10595e-5 new values x = 2.0000 y = 1.0000
#fsolve: new norm: 0.
#fsolve: iter = 6 |incr| = 0.31360e-12 new values x = 2.0000 y = 1.0000
#               {x = 2.000000000, y = 1.000000000}

Or using code from:https://www.mapleprimes.com/questions/200377-Nonlinear-Equations-With-Newoton-Newton#answer201481 thanks for Carl Love.

Download MultiNewton.mw

Download ILT_(1).mw

Maybe this helps:

0.https://mapleprimes.com/questions/223689-Comparing-RK4-And-RK45-How-To-Solve

1. https://www.mapleprimes.com/questions/223703-RungeKutta-4th-Order

2.https://www.mapleprimes.com/questions/221882-Maple-Code-For-4th-Order-Rungekutta

3.https://www.mapleprimes.com/questions/220556-How-Do-I-Implement-RungeKutta-Of-Order.

A test example:

Download testODE.mw

This package is build-in Maple,you don't need to download anything:

Execute:

with(Student[Basics]);

LinearSolveSteps((x+1)/(2*y*z) = 4*y^2/z+3*x/y, x);

LinearSolveSteps function have some limitation only  for linear equations give steps.

LinearSolveSteps("x^2 -2*x+1 = 0", x); #Nonlinear equation. It doesn't work.

Eliminating variable m1 and m2 from equations, then we have only eq. with 2 variables k1 and k2.

Ploting we see they do not intersect anywhere, even at point {0,0}.

Download No_solutions.mw

You must add 4 more (intial,boundary) condition if you what to solve by numeric method.

e.g. :{ U(0, t) = 1, U(1, t) = 0, V(0, t) = 0, V(1, t) = 0}

Executed in Maple 2018 !!!

Download pdesys.mw

Y := proc (delta, b, n) 
if n = 0 then 1/(b^2+1) 
elif n = 1 then arctan(1/b) 
elif n = 2 then 1+(1/2)*ln(b^2+1)-b*arctan(1/b) 
elif n = 3 then delta-(3/2)*b-(1/2)*b*ln(b^2+1)+(1/2)*arctan(1/b)*(b^2-1) 
elif n = 4 then (1/2)*delta^2-b*delta+(11/12)*b^2-11/36+(1/12)*ln(b^2+1)*(3*b^2-1)+(1/6)*b*arctan(1/b)*(3-b^2) 
elif n = 5 then (1/3)*delta^3-(1/2)*delta^2*b+(1/6)*delta*(3*b^2-1)+(25/72)*b-(25/72)*b^3+(1/12)*b*log(b^2+1)*(1-b^2)+(1/24)*arctan(1/b)*(1-6*b^2+b^4) 
elif n = 6 then (1/4)*delta^4-(1/3)*delta^2*b+(1/12)*delta^2*(3*b^2-1)+(1/6)*b*delta*(1-b^2)+(137/1440)*b^4-(137/720)*b^2+137/7200+(1/240)*log(b^2+1)*(5*b^4-10*b^2+1)+(1/120)*b*arctan(1/b)*(-5+10*b^2-b^4) 
elif n = 7 then (1/5)*delta^5-(1/4)*delta^4*b+(1/18)*delta^3*(3*b^2-1)+(1/12)*delta^2*b*(1-b^2)+(1/120)*delta*(5*b^4-10*b^2+1)-(49/2400)*b+(49/720)*b^3-(49/2400)*b^5+(1/720)*b*log(b^2+1)*(-3*b^4+10*b^2-3)+(1/720)*arctan(1/b)*(-1+15*b^2-15*b^4+b^6) 
end if 
end proc:

plot(Y(5, b, 7), b = 0 .. 10); # Example of use

u1 := proc (x, y) options operator, arrow; 1-exp(a*x)*cos(2*Pi*y) end proc;

a := -0.39323780;

evalf(int(u1(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#5.493248990

#OR:

u := unapply(1-exp(a*x)*cos(2*Pi*y), [x, y]);

a := -0.39323780;

evalf(int(u(x, y)^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

#OR:

u := 1-exp(a*x)*cos(2*Pi*y);

a := -0.39323780;

evalf(int(u^2, y = -.5 .. 1.5, x = -.5 .. 1.5));

Use  ExcelTools for import file.

Download testxlsx.mw

I don't have Maple V (This is an obsolete version,you need upgrade).

integral := Int(2*(sin(theta)/cos(theta))^(2*p-1), theta = 0 .. (1/2)*Pi);

with(IntegrationTools):

integral2 := Change(convert(integral, tan), tan(theta) = sqrt(t));

`assuming`([value(integral2)], [0 < p and p < 1]);

#Answer is: Pi*csc(Pi*p)

integral.mw

Executed in Maple 2018

Maple can't simplify better, probably V is not a solution of pde[1].

See attached file:

solution_ver_3.mw

BVP := [4*(diff(u(x, t), t))-9*(diff(u(x, t), x, x))-5*u(x, t) = 0, u(0, t) = 0, u(6, t) = 0, u(x, 0) = sin((1/6)*Pi*x)^2];

sol := pdsolve(BVP);

plot3d(eval(rhs(sol), infinity = 10), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, rhs(sol)), x = 0 .. 6, t = 0 .. 4);

OR:

plot3d(subs(infinity = 10, op(2, sol)), x = 0 .. 6, t = 0 .. 4)

 I am no expert in special functions,but adding comand "_EnvLegendreCut"

_EnvLegendreCut := 1 .. infinity;

plot(LegendreQ((1/2)*sqrt(5)-1/2, x), x = -1 .. 1);

for more info execute command:

?LegendreQ;

restart;

with(ListTools):

l := seq(n, n = -10 .. 20);

[l];

l1 := Flatten([Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]])]);

l2 := Transpose([[seq(n, n = -10 .. 20)], [seq(n, n = 1 .. 31)]]);

l3 := select(type, [seq(n, n = 115 .. 231)], 'odd');

[l3];

In yours first example Maple gives incorect answer.It's a Bug.!!!

evalf[10](sum(2^n*floor(2^n), n = 1 .. infinity));

#-1.333333333

evalf[10](Sum(2^n*floor(2^n), n = 1 .. infinity));# Big 'S' in Sum

#-1.333333333

`assuming`([sum(2^n*floor(2^n), n = 1 .. m)], [m > 0]);

#(1/3)*2^(m+1)*floor(2^(m+1))-4/3

limit(%, m = infinity)

#infinity

Second one give a correct answer:

evalf[10](sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));

#Float(infinity)

evalf[10](Sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. infinity));# Big 'S' in Sum

#If Maple dosen't know the answer then: Returns unevaluated,not infinity in this case.

`assuming`([sum(5^(n-1)*floor((1/4)*5^n), n = 1 .. m)], [m > 0]);

#Returns unevaluated

Executed in Maple 2018.