ODE
\[ (x-y(x)) \sqrt {y'(x)}=a \left (y'(x)+1\right ) \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 47.1926 (sec), leaf count = 51
\[\left \{\left \{y(x)\to -\frac {a^2+c_1 \left (c_1+x\right )}{c_1+x}\right \},\left \{y(x)\to -\frac {a^2-c_1 x+c_1^2}{x-c_1}\right \}\right \}\]
Maple ✓
cpu = 0.023 (sec), leaf count = 37
\[ \left \{ y \left ( x \right ) =x-2\,a,[x \left ( {\it \_T} \right ) ={a{\frac {1}{\sqrt {{\it \_T}}}}}+{\it \_C1},y \left ( {\it \_T} \right ) ={1 \left ( {\it \_C1}\,\sqrt {{\it \_T}}-{\it \_T}\,a \right ) {\frac {1}{\sqrt {{\it \_T}}}}}] \right \} \] Mathematica raw input
DSolve[(x - y[x])*Sqrt[y'[x]] == a*(1 + y'[x]),y[x],x]
Mathematica raw output
{{y[x] -> -((a^2 + C[1]*(x + C[1]))/(x + C[1]))}, {y[x] -> -((a^2 - x*C[1] + C[1]^2)/(x - C[1]))}}
Maple raw input
dsolve((x-y(x))*diff(y(x),x)^(1/2) = a*(1+diff(y(x),x)), y(x),'implicit')
Maple raw output
y(x) = x-2*a, [x(_T) = a/_T^(1/2)+_C1, y(_T) = (_C1*_T^(1/2)-_T*a)/_T^(1/2)]