##### 4.15.29 $$\sqrt {x y(x)} y'(x)-y(x)+x=\sqrt {x y(x)}$$

ODE
$\sqrt {x y(x)} y'(x)-y(x)+x=\sqrt {x y(x)}$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.366606 (sec), leaf count = 61

$\text {Solve}\left [\frac {1}{1-\sqrt {\frac {y(x)}{x}}}+\frac {3}{2} \log \left (1-\sqrt {\frac {y(x)}{x}}\right )+\frac {1}{2} \log \left (\sqrt {\frac {y(x)}{x}}+1\right )+\log (x)=c_1,y(x)\right ]$

Maple
cpu = 0.075 (sec), leaf count = 48

$\left [\frac {\ln \left (\sqrt {x y \left (x \right )}+x \right )}{3}+\ln \left (-x +\sqrt {x y \left (x \right )}\right )-\frac {2 x}{3 \left (-x +\sqrt {x y \left (x \right )}\right )}-\frac {2 \ln \left (x \right )}{3}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[x - y[x] + Sqrt[x*y[x]]*y'[x] == Sqrt[x*y[x]],y[x],x]

Mathematica raw output

Solve[Log[x] + (3*Log[1 - Sqrt[y[x]/x]])/2 + Log[1 + Sqrt[y[x]/x]]/2 + (1 - Sqrt
[y[x]/x])^(-1) == C[1], y[x]]

Maple raw input

dsolve(diff(y(x),x)*(x*y(x))^(1/2)+x-y(x) = (x*y(x))^(1/2), y(x))

Maple raw output

[1/3*ln((x*y(x))^(1/2)+x)+ln(-x+(x*y(x))^(1/2))-2/3*x/(-x+(x*y(x))^(1/2))-2/3*ln
(x)-_C1 = 0]