ODE
\[ 2 \sqrt {a y'(x)}+x y'(x)-y(x)=0 \] ODE Classification
[[_homogeneous, `class G`], _Clairaut]
Book solution method
Clairaut’s equation and related types, main form
Mathematica ✓
cpu = 91.4646 (sec), leaf count = 93
\[\left \{\left \{y(x)\to \left (\cosh \left (2 c_1\right )-\sinh \left (2 c_1\right )\right ) \left (-\left (2 i \sqrt {a} \sinh \left (c_1\right )+2 i \sqrt {a} \cosh \left (c_1\right )+x\right )\right )\right \},\left \{y(x)\to i \left (\cosh \left (2 c_1\right )-\sinh \left (2 c_1\right )\right ) \left (2 \sqrt {a} \sinh \left (c_1\right )+2 \sqrt {a} \cosh \left (c_1\right )+i x\right )\right \}\right \}\]
Maple ✓
cpu = 0.024 (sec), leaf count = 15
\[ \left \{ y \left ( x \right ) =2\,\sqrt {{\it \_C1}\,a}+{\it \_C1}\,x \right \} \] Mathematica raw input
DSolve[-y[x] + x*y'[x] + 2*Sqrt[a*y'[x]] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((x + (2*I)*Sqrt[a]*Cosh[C[1]] + (2*I)*Sqrt[a]*Sinh[C[1]])*(Cosh[2*C[1]] - Sinh[2*C[1]]))}, {y[x] -> I*(I*x + 2*Sqrt[a]*Cosh[C[1]] + 2*Sqrt[a]*Sinh[C[1]])*(Cosh[2*C[1]] - Sinh[2*C[1]])}}
Maple raw input
dsolve(2*(a*diff(y(x),x))^(1/2)+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = 2*(_C1*a)^(1/2)+_C1*x