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

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

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

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

Mathematica
cpu = 0.89479 (sec), leaf count = 201

\[\left \{\text {Solve}\left [\frac {x \left (-1-i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}\right )+2 y(x) \left (\log \left (\frac {y(x)}{x}\right )-i \tan ^{-1}\left (\frac {x \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}}{y(x)+x}\right )+\log (x)\right )}{4 y(x)}=c_1,y(x)\right ],\text {Solve}\left [\frac {x \left (-1+i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}\right )+2 y(x) \left (\log \left (\frac {y(x)}{x}\right )+i \tan ^{-1}\left (\frac {x \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}}{y(x)+x}\right )+\log (x)\right )}{4 y(x)}=c_1,y(x)\right ]\right \}\]

Maple
cpu = 0.085 (sec), leaf count = 243

\[ \left \{ {\frac {1}{2\,xy \relax (x ) } \left (-{x}^{2} \left ({\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}} \right ) ^{{\frac {3}{2}}}+ \left (2\,xy \relax (x ) -3\, \left (y \relax (x ) \right ) ^{2} \right ) \sqrt {{\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}}}-2\,x \left (-\ln \left ({\frac {y \relax (x ) }{x}} \right ) y \relax (x ) + \left ({\it Artanh} \left ({\frac {x+y \relax (x ) }{x}{\frac {1}{\sqrt {{\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}}}}}} \right ) +{\it \_C1}-\ln \relax (x ) \right ) y \relax (x ) +x/2 \right ) \right ) }=0,{\frac {1}{2\,xy \relax (x ) } \left ({x}^{2} \left ({\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}} \right ) ^{{\frac {3}{2}}}+ \left (-2\,xy \relax (x ) +3\, \left (y \relax (x ) \right ) ^{2} \right ) \sqrt {{\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}}}+2\,x \left (\ln \left ({\frac {y \relax (x ) }{x}} \right ) y \relax (x ) + \left ({\it Artanh} \left ({\frac {x+y \relax (x ) }{x}{\frac {1}{\sqrt {{\frac { \left (x+3\,y \relax (x ) \right ) \left (x-y \relax (x ) \right ) }{{x}^{2}}}}}}} \right ) -{\it \_C1}+\ln \relax (x ) \right ) y \relax (x ) -x/2 \right ) \right ) }=0,y \relax (x ) =x \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[(2*((-I)*ArcTan[(x*Sqrt[-1 + y[x]/x]*Sqrt[1 + (3*y[x])/x])/(x + y[x])] + 
Log[x] + Log[y[x]/x])*y[x] + x*(-1 - I*Sqrt[-1 + y[x]/x]*Sqrt[1 + (3*y[x])/x]))/
(4*y[x]) == C[1], y[x]], Solve[(2*(I*ArcTan[(x*Sqrt[-1 + y[x]/x]*Sqrt[1 + (3*y[x
])/x])/(x + y[x])] + Log[x] + Log[y[x]/x])*y[x] + x*(-1 + I*Sqrt[-1 + y[x]/x]*Sq
rt[1 + (3*y[x])/x]))/(4*y[x]) == C[1], y[x]]}

Maple raw input

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

Maple raw output

y(x) = x, 1/2*(-x^2*((x+3*y(x))*(x-y(x))/x^2)^(3/2)+(2*x*y(x)-3*y(x)^2)*((x+3*y(
x))*(x-y(x))/x^2)^(1/2)-2*x*(-ln(y(x)/x)*y(x)+(arctanh((x+y(x))/x/((x+3*y(x))*(x
-y(x))/x^2)^(1/2))+_C1-ln(x))*y(x)+1/2*x))/x/y(x) = 0, 1/2*(x^2*((x+3*y(x))*(x-y
(x))/x^2)^(3/2)+(-2*x*y(x)+3*y(x)^2)*((x+3*y(x))*(x-y(x))/x^2)^(1/2)+2*x*(ln(y(x
)/x)*y(x)+(arctanh((x+y(x))/x/((x+3*y(x))*(x-y(x))/x^2)^(1/2))-_C1+ln(x))*y(x)-1
/2*x))/x/y(x) = 0