TITLE: How find this equation integer solution:$x^2y^2=4x^5+y^3$
QUESTION [10 upvotes]: find $x,y\in Z$,and such that $$x^2y^2=4x^5+y^3$$
and I use mathmatical give me:

my try:
Let $gcd(x,y)=d$ and put $x=da, y=db$, with $a$ and $b$ coprime:
$d^4a^2b^2 = 4d^5a^5 + d^3b^3 \Rightarrow da^2b^2=4d^2a^5+b^3$. Hence, we have that $a|b^3$ and as $gcd(a,b)=1$, $a=1$ or $a=-1$. Plugging it back, we have $4d^2-db^2+b^3=0$ or $4d^2+db^2-b^3=0$. 
The discriminant, in the first case, is $b^4-16b^3$, which must be a perfect square. Therefore, $b^2-16b = k^2$, where $k$ is an integer. This gives us $(b-8)^2 - k^2 = 64$.
From this, we get the following pairs: $(125,3025), (27, 486), (54, 972), (32, 512)$.
In the second case, the discriminant is $b^4+16b^3$, which must be a perfect square again. Therefore, $b^2+16b=l^2$, where $l$ is an integer. This gives us $(b+8)^2-l^2 = 64$.
From this, we get the following pairs: $(0,0), (2,-4), (-1,2), (27, -243)$
my try is true? and have other methods?

REPLY [3 votes]: The basic equation is defined here for real $(x,y)$ as well, by: 
$$
   x^2 y^2 = 4 x^5 + y^3
$$
See picture ; window $-200 < x < +200$ , $-4000 < y < +4000$ .
It is immediately clear that $(x,y) = (0,0)$ is an (integer)
solution of the equation and that $x = 0 \leftrightarrow y = 0$ .
$\color{red}{Tangents}\,$ at the curve are calculated by implicit differentation:
$$
  2 x y^2 + x^2 2 y y' = 20 x^4 + 3 y^2 y' \quad \Longrightarrow \quad
  y' = \frac{20 x^4 - 2 x y^2}{2 y x^2 - 3 y^2}
$$
Horizontal tangents at:
$$
   20 x^4 - 2 x y^2 = 0 \quad \Longrightarrow \quad y^2 = 10 x^3
$$
Substitute into the basic equation to obtain:
$$
   x^2 y^2 = 4 x^5 + y^3 \quad \Longrightarrow \quad
   x^2 \cdot 10 x^3 = 4 x^5 + 10 x^3 y
$$
We have covered the trivial solution $(0,0)$ so forget about it in the sequel.
$$
   6 x^2 = 10 y \quad \Longrightarrow \quad 36 x^4 = 100 y^2 = 1000 x^3
   \quad \Longrightarrow \quad (x_S,y_S) = \left(\frac{10^3}{6^2},\sqrt{\frac{10^{10}}{6^6}}\right)
$$
The Special point  S  is important for the numerical work
that follows. Vertical tangents at:
$$
   2 y x^2 - 3 y^2 = 0 \quad \Longrightarrow \quad y = \frac{2}{3} x^2
$$
Substitute into the basic equation to obtain:
$$
   x^2 \cdot \frac{4}{9} x^4 = 4 x^5 + \frac{8}{27} x^6
   \quad \Longrightarrow \quad \frac{4}{27} x = 4
   \quad \Longrightarrow \quad (x,y) = (27,486)
$$
The latter happens to be one of the required integer solutions !The rest of the method is brute force.
It is clear from the picture that there are four branches:

one extending from $(0,0)$ towards minus infinity
    in the quadrant $x > 0$ , $y < 0$indicated as A
one extending from $(0,0)$ towards plus infinity
    in the quadrant $x < 0$ , $y > 0$indicated as B
one extending from the special point  S  towards plus infinity
    on the right sidein the quadrant $x > 0$ , $y > 0$ , indicated as C
one extending from the special point  S  towards plus infinity
    on the left sidein the quadrant $x > 0$ , $y > 0$ , indicated as D

Therefore four "crawlers" are contained in the computer program, which is
listed below.The crawlers seek for integer solutions by crawling along the
four branches of the curve, for some time, until someone hits the
Ctrl-C key to halt. It's clearly inferior to the method presented by the other author (: Neil), but it finds all the solutions, except some trivial ones.

program krabbel;
function pow(x : double; n : integer) : double;
var
  q : double;
  k : integer;
begin
  q := 1;
  for k := 1 to n do
    q := q * x;
  pow := q;
end;
function f(x,y : double) : double;
begin
  f := pow(x,2)*pow(y,2) - (4*pow(x,5) + pow(y,3));
end;
procedure crawl_A;
var
  x,y : integer;
  g : double;
begin
  x := 0; y := 0;
  while true do
  begin
    y := y - 1;
    g := f(x,y);
    if g = 0 then Writeln(x,' ',y);
    while g > 0 do
    begin
      x := x + 1;
      g := f(x,y);
      if g = 0 then Writeln(x,' ',y);
    end;
  end;
end;
procedure crawl_B;
var
  x,y : integer;
  g : double;
begin
  x := 0; y := 0;
  while true do
  begin
    y := y + 1;
    g := f(x,y);
    if g = 0 then Writeln(x,' ',y);
    while g < 0 do
    begin
      x := x - 1;
      g := f(x,y);
      if g = 0 then Writeln(x,' ',y);
    end;
  end;
end;
procedure crawl_C;
var
  x,y : integer;
  g : double;
begin
  x := Round(pow(10,3)/pow(6,2));
  y := Round(sqrt(pow(10,10)/pow(6,6)));
  while true do
  begin
    y := y + 1;
    g := f(x,y);
    if g = 0 then Writeln(x,' ',y);
    while g > 0 do
    begin
      x := x + 1;
      g := f(x,y);
      if g = 0 then Writeln(x,' ',y);
    end;
  end;
end;
procedure crawl_D;
var
  x,y : integer;
  g : double;
begin
  x := 27; y := 486;
  while true do
  begin
    y := y + 1;
    g := f(x,y);
    if g = 0 then Writeln(x,' ',y);
    while g < 0 do
    begin
      x := x + 1;
      g := f(x,y);
      if g = 0 then Writeln(x,' ',y);
    end;
  end;
end;
begin
{ crawl_A; crawl_B; } crawl_C; { crawl_D; }
end.

Results $\;\rightarrow picture: \color{red}{dots}$

by hand:
0 0
27 486
crawl_A:
2 -4
27 -243
crawl_B:
-1 2
crawl_C:
32 512
54 972
125 3125
crawl_D:
nothing