I have just explained that.

@Markiyan Hirnyk 

Actually the infinity of solutions refers to ode=0, where:

ode:=(1/2)*(diff(y(x), x))^2 - (1-ln(y(x)^2))*y(x)^2;

This should be normal because the  solve(...)  was used and it gives 2 solutions.

Ya verifies this ode:

simplify( eval(ode, y(x)=Ya) );
    0

If we restrict to solve()[1]  then we have only two solutions.

I don't see a bug here. Your ode (y' = sqrt(2-2*ln(y^2))*y, y(0)=1)  has infinitely many solutions in the interval [0,1]. Two of them are in the plot.

@acer 

evalf/int/control is using a scaling factor of 10^-280 (!?).
I would expect evalf/Int to make use of a minimal amount of symbolic computations but instead it does lots of symbolic with evala.

@Axel Vogt 

Same for me for Digits:=15; but for Digits:=10 the crash is fast (mserver.exe).

@acer 

Yes, and doing so, Q1 will be orthogonal!

@tomleslie 

But the two systems are incompatible. Compare the second equations of each.
==> r(x)^2 * T1(x) = 0 ==> r(x)=0  (because T1(x) <>0).

@Aaeru Michi 

As you see in the practice example, Maple does not accept an integro-differential system of this kind.
You must transform it into ODEs, and it's simple; why do you reject this? You have no other choice.

I don't understand your last system.  If you mean:
y'(t)=x(t)^2+z(t)
x''(t)=y(t)+t
z(t)=int ( x(t1)*y(t1), t1=0 .. t )

then you must transform it as above: z' = x*y, z(0)=0.

@_Maxim_ 

Scaling is not a problem when solve/series works. The bottleneck is solve itself.
Try e.g. the similar  F := (x, y) -> ln((1+x)*y)+exp(x^2+y-1)-x-cos(x),  F(0,1)=0. ( D[2](F)(0,1) = 2 <> 0. )

p:=ln((1+x)*y) + exp((x^2)*(y^2)) - x - cos(x):
Order:=10:
'ytaylor'=solve(series(p,x),y);
ytaylor:=convert(solve(series(p,x,5),y), polynom): # useful only near 0
with(plots):
ip:=implicitplot(p, x=0..1, y=0..3, color=green):
display(plot(ytaylor,x=0..1,color=red), ip);

@Math Pi Euler 

p:=ln((1+x)*y) + exp((x^2)*(y^2)) - x - cos(x):
solve(series(p,x),y);

@Markiyan Hirnyk 

OK, thank you; it seems that only Maple on 64 bit is affected.

@Markiyan Hirnyk 

Exporting in pdf works but it's a low resolution bitmap.

Compare with a Maxima plot:
max.pdf

The dropbox .eps  was it from classic or standard interface?
[for 64 bit, the classic interface is absent].

A 2D .pdf can be obtained exporting in .eps format (encapsulated postscript) and then converting into .pdf using Ghostscript.
It results a nice vector graphic.
Note that on my system, exporting directly in .pdf (with the context menu) it results a low resolution bitmap.

The big problem is that the .eps is totally broken for 3d (for many years!). Practically, Maple does not have 3d vector graphics!

p:=plot3d(x^2+y^2, x=-1..1,y=-1..1):
plottools:-exportplot("v3d.eps",p);