4.10.40 \(4 (-y(x)-x+1) y'(x)-x+2=0\)

ODE
\[ 4 (-y(x)-x+1) y'(x)-x+2=0 \] ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class C`], _dAlembert]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 4.83265 (sec), leaf count = 109

\[\text {Solve}\left [\frac {2^{2/3} \left (x \log \left (\frac {x-2}{y(x)+x-1}\right )-x \log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+2 y(x) \left (\log \left (\frac {x-2}{y(x)+x-1}\right )-\log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+1\right )+2 x-2\right )}{9 (2 y(x)+x)}=c_1+\frac {1}{9} 2^{2/3} \log (x-2),y(x)\right ]\]

Maple
cpu = 0.023 (sec), leaf count = 59

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

DSolve[2 - x + 4*(1 - x - y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(2^(2/3)*(-2 + 2*x + x*Log[(-2 + x)/(-1 + x + y[x])] - x*Log[(x + 2*y[x])/
(-1 + x + y[x])] + 2*(1 + Log[(-2 + x)/(-1 + x + y[x])] - Log[(x + 2*y[x])/(-1 +
 x + y[x])])*y[x]))/(9*(x + 2*y[x])) == C[1] + (2^(2/3)*Log[-2 + x])/9, y[x]]

Maple raw input

dsolve(4*(1-x-y(x))*diff(y(x),x)+2-x = 0, y(x),'implicit')

Maple raw output

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