On integer solutions to $x\sqrt{y}+y\sqrt{x} = a, x+y = b $ A question in quora
asked to 
find real solution(s) to
$x\sqrt{y}+y\sqrt{x} = 6,
x+y = 5.
$
I showed that the solution
with $x \le y$ is
$x = 1, y = 4
$.
This naturally
brings up the question
for 
which positive integers $a$ and $b$
do the equations
$x\sqrt{y}+y\sqrt{x} = a,
x+y = b
$
have integer $x$ and $y$
as solutions.
It is clear that
for any integers
$1 \le p \le q$,
$a = pq(p+q), b = p^2+q^2$
has the solution
(with $x \le y$) of
$x = p^2, y = q^2$.
The original question has
$p=1, q=2$.
My question is:
are there any other
integral $a$ and $b$
for which the
equations have
integral solutions?
Note:
If we just
try to solve for $x$,
this happens
(with the help of Wolfy):
$\begin{array}\\
y 
&= b-x\\
a
&=x\sqrt{y}+y\sqrt{x}\\
&=x\sqrt{b-x}+(b-x)\sqrt{x}\\
a-x\sqrt{b-x}
&=(b-x)\sqrt{x}\\
a^2-2ax\sqrt{b-x}+x^2(b-x)
&=x(b-x)^2\\
&=x(b^2-2bx+x^2)\\
2ax\sqrt{b-x}
&=x^3-2bx^2+b^2x
-a^2-(bx^2-x^3)\\
&=2x^3-3bx^2+b^2x-a^2\\
4a^2x^2(b-x)
&=(2x^3-3bx^2+b^2x-a^2)^2\\
0
&=a^4 - 2 a^2 b^2 x + 2 a^2 b x^2 + b^4 x^2 - 6 b^3 x^3 + 13 b^2 x^4 - 12 b x^5 + 4 x^6\\
\text{with real roots}\\
x 
&= \dfrac12 \left(b \pm \sqrt{b^2 - 4 \left(-\dfrac{r}{12} + \dfrac{(24 a^2 b - b^4)}{12 r} + \dfrac{b^2}{12}\right)}\right)\\
\text{where}\\
r
&=\left(-216 a^4 + 36 a^2 b^3 + 24 \sqrt{3} \sqrt{27 a^8 - a^6 b^3} - b^6\right)^{1/3}\\
\end{array}
$
I don't know
how much help
this is.

If $$x\sqrt{y}+y\sqrt{x} = a,\tag{1}$$ with $x,y,a\in \mathbb{N},$ then $$x^2y+2xy\sqrt{xy}+xy^2=a^2,$$ so that $\sqrt{xy}$ is rational, and $xy$ is a perfect square.  Let $g=\gcd(x,y),$ so that $x=gm, y=gn$ where $m$ and $n$ are co-prime integers.  We have that $\sqrt{nm}$ is rational, so that $nm$ is a perfect square, and since $\gcd(n,m)=1$ $m$ and $n$ are each perfect squares, say $x=gr^2,\ y=gs^2,\ r,s\in\mathbb{N}$.  
Substituting these values into $(1)$ we get that $\sqrt{g}$ is rational, so that $g$ is a perfect square, and finally, that $x$ and $y$ are perfect squares.
In short, you have found all the possibilities. 
