I found a workaround. See attached file.

DIFF_eq.mw

Download PrimePi_-approximation.mw

I have analitycal solution only with conditions:

[f3(0,t)=-exp(-2*t),f4(x,0)=exp(-3*x)*cos(2*Pi*x),f1(x,3)=0,f1(3,t)=0,f2(3,0)=0].

last condition You may change in the code from :f2(3,0)=0 to f2(0,3)=0

from eval(rhs(sol[2]), x = 3) = 0 to eval(rhs(sol[2]), t = 3) = 0 in SOL.

See attached file.

EQ_diff.mw

I don't have Maple 13.I used Maple 2019.2:

SOL := solve([13/4*m - 7/4*n - 3 = 0, -17/2*n*2^n + 34*m = 0]);

evalf(map(allvalues, {SOL}));

Gives:

{{m = -0.1262032999, n = -1.948663272}, {m = 2., n = 2.}, {m = 1.551915566 - 4.737602358*I, n = 1.167843193 - 8.798404378*I}, {m = 1.551915566 + 4.737602358*I, n = 1.167843193 + 8.798404378*I}}

I have no more adequate knowledge about multithreading in Maple,but my code seems to work. 

time give you total CPU time,I used  packages CodeTools to give You a REAL time computation.

Download thread_time_AAA.mw

Using assumption and infinity not Infinity, works fine.

Integrals_Sheet2-A.mw

One way is using this package.

Extract file aene_v1_0.tar to any folder then open a file Sample.mws and copy my code to the file and execute.

eq:=diff(y(t),t$mu)=0.85*sqrt(y(t)^2)-0.85*y(t)^2:
IC:=[y(0)=0.5]:
Asolve([eq],IC,mu=1/2,t=0..1,y=0..1,false,index=10,output=plot,pade=[10,10]);

Or workaround:

restart;
T__ln := (1/2)*m__ws*v__l+(1/2)*I__ws*(diff(varphi__l(t), t))^2+((1/2*(m__zs+m__zp+m__wp))*v__l+(1/2*(I__zs+I__zp+I__wp))*(diff(varphi__l(t), t))^2)*z__11*eta/z__12+((1/2*(m__wk+m__zpk+m__k+m__zk))*v__l+(1/2*(I__wk+I__zpk+I__k+I__zk))*(diff(varphi__l(t), t))^2)*z__21*eta/z__22;
eval(diff(eval(T__ln, diff(varphi__l(t), t) = x), x), x = diff(varphi__l(t), t));

With package: MultiSeries Maple can:

N := 10;
(MultiSeries:-series(sin(x)^a, x = 0, N) assuming (0 < a));

#x^a - a*x^(2 + a)/6 + (-1/180*a + 1/72*a^2)*x^(4 + a) + (-1/2835*a + 1/1080*a^2 - 1/1296*a^3)*x^(6 + a) + (-1/37800*a + 101/1360800*a^2 - 1/12960*a^3 + 1/31104*a^4)*x^(8 + a) + O(x^(10 + a))

Another simpler way to compute this integral:

Download Compute_integral.mw

A good question why inttrans does not work in this case? I don't know.

A trick: 

inttrans:-fourier(- diff(erf(x), x)*I/k, x, k);

# -2*I*exp(-k^2/4)/k # A correct answer.

int(erf(x)*exp(-I*x*k),x=-infinity..infinity);

# -2*I*exp(-k^2/4)/k !

evalf[10](Int(exp(-x)*sin(x^2/2)/(3 + x), x = 1 .. infinity));

#0.04279639331

Or:

Digits := 10:

int(exp(-x)*sin(x^2/2)/(3 + x), x = 1 .. infinity, numeric);

#0.04279639331

Execute: ?Digits ,to see more information.

with Maple 2019.2 :

int(1/(c*x + d), x = a .. b) assuming a < b;

#piecewise(And(a < -d/c, -d/c < b), undefined, (-ln(a*c + d) + ln(b*c + d))/c)

Do you have any reason to think there is a closed form? Most nonlinear ODE's don't have one.

With numerics,see attached file:
 

Download ODE.mw

You have a BVP type ODE ,RK methods dosen't work(in Maple) for yours ICS and BC.

Download BVP.mw