4.15.29 $\sqrt{xy(x)}{y}^{\prime }(x)-y(x)+x=\sqrt{xy(x)}$

ODE
$$\sqrt{xy(x)}{y}^{\prime }(x)-y(x)+x=\sqrt{xy(x)}$$ ODE Classification

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

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0872606 (sec), leaf count = 60

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

Maple ✓
cpu = 0.071 (sec), leaf count = 50

$$\{\frac{1}{3}\ln (\sqrt{xy\left(x\right)}+x)+2\frac{x}{3x-3\sqrt{xy\left(x\right)}}+\ln (-x+\sqrt{xy\left(x\right)})-\frac{2\ln \left(x\right)}{3}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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]] + Log[1 + Sqrt[y[x]/x]] - 2/(-1 + Sqrt[y
[x]/x]))/2 == 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),'implicit')

Maple raw output

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