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.

Using CauchyPrincipalValue =true:

`assuming`([int(exp(I*x*t)/((x-a)*(x+a)), x = -infinity .. infinity, CauchyPrincipalValue = true)], [a > 0, t > 0])

#(1/2*I)*Pi*(exp(I*a*t)-exp(-I*a*t))/a

int.mw

Exectuted in Maple 2018.

remove[flatten](x-> x = 0, [seq(sin((1/4)*k*Pi), k = 1 .. 8)]);

The function RootOf is a placeholder for representing all the roots of an equation in one variable.  In particular, it is the standard representation for Maple algebraic numbers, algebraic functions.

For more info execute this code in Maple:

?RootOf

sol := solve({x^2+y^2 = 3, x^2+2*y^2 = 3}, {x, y});
allvalues(sol);
evalf(%);

#{x = sqrt(3), y = 0}, {x = -sqrt(3), y = 0}
#{x = 1.732050808, y = 0.}, {x = -1.732050808, y = 0.}

I don't know way it's happens,but if we convert to rational(exact) numbers then works:

eq := [190.0315457*d^6-7.601261828*d^4+344.0*c^2*d^2-2.677685950*c^4, 0.6830134554e-2*c^4+2084.317025*d^8-166.7453620*d^6+3.334907240*d^4-c^2*d^2];
eliminate(convert(eq, rational, exact), c);

#[{c = -(1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675-17435*sqrt(9667390500*d^2+561670891405))*d}, {121*d^4*(582104796124362625620*d^4-46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)-1272306087125807465)}], [{c = -(1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}], [{c = (1/14265)*sqrt(13071106675+17435*sqrt(9667390500*d^2+561670891405))*d}, {-121*d^4*(-582104796124362625620*d^4+46433010457684955790*d^2+2932165094782*sqrt(9667390500*d^2+561670891405)+1272306087125807465)}]

Download Latex_Sum.mw

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = `+`);

#A simple solutions.

pdsolve([diff(f2(lam, mu, l, m), m)-(diff(f1(lam, mu, l, m), l))-(diff(f4(lam, mu, l, m), mu))+diff(f3(lam, mu, l, m), lam) = 0, diff(f1(lam, mu, l, m), m)+diff(f2(lam, mu, l, m), l)-(diff(f3(lam, mu, l, m), mu))-(diff(f4(lam, mu, l, m), lam)) = 0], HINT = TWS(sin));

#Not simple solution.For more info. execute this code below?

?pdsolve

Probably you want:

with(Fractals:-LSystem); with(LSystemExamples):

PlotExample(DragonCurve, 15);

Lindenmayer System Plot Generator:

states := "FX";

rules := ["Y" = "FX-Y", "X" = "X+YF"];

cons := ["F" = "draw:1", "+" = "turn:-90", "-" = "turn:90"];

newstate1 := Iterate(states, rules, 10);

LSystemPlot(newstate1, cons);

EDITED:

Code from book Geometry of Curves and Surfaces with MAPLE. https://books.google.pl/books/about/Geometry_of_Curves_and_Surfaces_with_MAP.html?id=78w0CseXgvMC&redir_esc=y

dragon := proc (k::algebraic, N::integer) local t, i, q1, q2, q3, q4, d; global p; q2 := [k, 0]; q3 := [1-k, 0]; d := evalm(q3-q2); p[0] := plot([[q2[1], q2[2]], [q3[1], q3[2]]]); for i to N do if `mod`(i, 2) = 0 then t[i] := t[(1/2)*i] else t[i] := (`mod`(i, 4))-2 end if; q4 := evalm(q3+k*d/(1-2*k)); d := evalm(t[i]*[d[2], -d[1]]); q1 := evalm(q3); q2 := evalm(q4+k*d/(1-2*k)); q3 := evalm(q2+d); p[i] := plot([[q1[1], q1[2]], [q2[1], q2[2]], [q3[1], q3[2]]]) end do; return plots:-display([seq(p[i], i = 0 .. N)]) end proc;

dragon(0.1, 500);

f := x^2*y^2+3*x*y^2;

fdiff(f, [x, y], {x = 1, y = 2});

See in Help for more information,exectute code below:

?fdiff 

restart;

de := diff(y(t), t, t) = -1; ic := y(0) = 1, (D(y))(0) = 0;

Events := [y(t), diff(y(t), t) = -.5*(diff(y(t), t))];

Events2 := [y(t) = -2*(1/1000), halt];

dsol := dsolve({de, ic}, numeric, events = [Events, Events2], range = 0 .. 5);

plots[odeplot](dsol, thickness = 3, color = red)

You can do it numerically:

c[1]:=1;
c[2]:=1;
int(12.*x^3*c[2]+6.*x^2*c[1]+x^2*exp(x^3*c[2])*exp(x^2*c[1]), x = 0. .. 1.,numeric);

#6.155281446

If you looking a analyticaly solution it's probably not possible.

See :https://en.wikipedia.org/wiki/Integral#Symbolic