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 = 0.0872606 (sec), leaf count = 60

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

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

\[ \left \{ {\frac {1}{3}\ln \left (\sqrt {xy \relax (x ) }+x \right ) }+2\,{\frac {x}{3\,x-3\,\sqrt {xy \relax (x ) }}}+\ln \left (-x+\sqrt {xy \relax (x ) } \right ) -{\frac {2\,\ln \relax (x ) }{3}}-{\it \_C1}=0 \right \} \] 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]] + 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