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

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

Book solution method
Clairaut’s equation and related types, main form

Mathematica
cpu = 0.774037 (sec), leaf count = 167

\[\left \{\left \{y(x)\to \frac {1}{16} \left (-4 \sqrt {-a e^{2 c_1} (x+1)^2}-4 a (2 x+1)+e^{2 c_1}\right )\right \},\left \{y(x)\to -\sqrt {-a e^{2 c_1} (x+1)^2}-\frac {1}{4} a (2 x+1)+e^{2 c_1}\right \},\left \{y(x)\to \frac {1}{16} \left (4 \sqrt {-a e^{2 c_1} (x+1)^2}-4 a (2 x+1)+e^{2 c_1}\right )\right \},\left \{y(x)\to \sqrt {-a e^{2 c_1} (x+1)^2}-\frac {1}{4} a (2 x+1)+e^{2 c_1}\right \}\right \}\]

Maple
cpu = 0.019 (sec), leaf count = 25

\[ \left \{ y \relax (x ) ={\frac {{\it \_C1}\, \left (ax-{\it \_C1} \right ) }{a}},y \relax (x ) ={\frac {a{x}^{2}}{4}} \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^(2*C[1]) - 4*Sqrt[-(a*E^(2*C[1])*(1 + x)^2)] - 4*a*(1 + 2*x))/16}, 
{y[x] -> E^(2*C[1]) - Sqrt[-(a*E^(2*C[1])*(1 + x)^2)] - (a*(1 + 2*x))/4}, {y[x] 
-> (E^(2*C[1]) + 4*Sqrt[-(a*E^(2*C[1])*(1 + x)^2)] - 4*a*(1 + 2*x))/16}, {y[x] -
> E^(2*C[1]) + Sqrt[-(a*E^(2*C[1])*(1 + x)^2)] - (a*(1 + 2*x))/4}}

Maple raw input

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

Maple raw output

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