4.19.5 \(-(a+2 x y(x)) y'(x)+x^2 y'(x)^2+y(x)^2=0\)

ODE
\[ -(a+2 x y(x)) y'(x)+x^2 y'(x)^2+y(x)^2=0 \] ODE Classification

[[_homogeneous, `class G`], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)

Mathematica
cpu = 0.273771 (sec), leaf count = 47

\[\left \{\left \{y(x)\to \frac {x-2 \sqrt {a} c_1}{4 c_1^2}\right \},\left \{y(x)\to \frac {2 \sqrt {a} c_1+x}{4 c_1^2}\right \}\right \}\]

Maple
cpu = 0.044 (sec), leaf count = 36

\[ \left \{ y \relax (x ) =-{\frac {a}{4\,x}},y \relax (x ) ={\it \_C1}\,x-\sqrt {a{\it \_C1}},y \relax (x ) ={\it \_C1}\,x+\sqrt {a{\it \_C1}} \right \} \] Mathematica raw input

DSolve[y[x]^2 - (a + 2*x*y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x - 2*Sqrt[a]*C[1])/(4*C[1]^2)}, {y[x] -> (x + 2*Sqrt[a]*C[1])/(4*C[1
]^2)}}

Maple raw input

dsolve(x^2*diff(y(x),x)^2-(a+2*x*y(x))*diff(y(x),x)+y(x)^2 = 0, y(x),'implicit')

Maple raw output

y(x) = -1/4/x*a, y(x) = _C1*x-(a*_C1)^(1/2), y(x) = _C1*x+(a*_C1)^(1/2)