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

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.175631 (sec), leaf count = 121

\[\left \{\left \{y(x)\to -\frac {2}{3} \sqrt {e^{c_1} \left (e^{c_1}-3 x\right )}-\frac {e^{c_1}}{3}\right \},\left \{y(x)\to \frac {2}{3} \sqrt {e^{c_1} \left (e^{c_1}-3 x\right )}-\frac {e^{c_1}}{3}\right \},\left \{y(x)\to e^{c_1}-2 \sqrt {e^{c_1} \left (e^{c_1}+x\right )}\right \},\left \{y(x)\to 2 \sqrt {e^{c_1} \left (e^{c_1}+x\right )}+e^{c_1}\right \}\right \}\]

Maple
cpu = 0.073 (sec), leaf count = 107

\[ \left \{ {x}^{2}+xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}=0,\ln \relax (x ) -{\it Artanh} \left ({\frac {2\,x+y \relax (x ) }{2\,x}{\frac {1}{\sqrt {{\frac {{x}^{2}+xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) +\ln \left ({\frac {y \relax (x ) }{x}} \right ) -{\it \_C1}=0,\ln \relax (x ) +{\it Artanh} \left ({\frac {2\,x+y \relax (x ) }{2\,x}{\frac {1}{\sqrt {{\frac {{x}^{2}+xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) +\ln \left ({\frac {y \relax (x ) }{x}} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -E^C[1]/3 - (2*Sqrt[E^C[1]*(E^C[1] - 3*x)])/3}, {y[x] -> -E^C[1]/3 + (
2*Sqrt[E^C[1]*(E^C[1] - 3*x)])/3}, {y[x] -> E^C[1] - 2*Sqrt[E^C[1]*(E^C[1] + x)]
}, {y[x] -> E^C[1] + 2*Sqrt[E^C[1]*(E^C[1] + x)]}}

Maple raw input

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

Maple raw output

x^2+x*y(x)+y(x)^2 = 0, ln(x)-arctanh(1/2*(2*x+y(x))/x/((x^2+x*y(x)+y(x)^2)/x^2)^
(1/2))+ln(y(x)/x)-_C1 = 0, ln(x)+arctanh(1/2*(2*x+y(x))/x/((x^2+x*y(x)+y(x)^2)/x
^2)^(1/2))+ln(y(x)/x)-_C1 = 0