4.38.3 \(x y''(x)=(1-y(x)) y'(x)\)

ODE
\[ x y''(x)=(1-y(x)) y'(x) \] ODE Classification

[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0717102 (sec), leaf count = 46

\[\left \{\left \{y(x)\to 2-\sqrt {2} \sqrt {c_1+2} \tanh \left (\frac {\sqrt {c_1+2} \left (2 c_2-\log (x)\right )}{\sqrt {2}}\right )\right \}\right \}\]

Maple ✓
cpu = 0.314 (sec), leaf count = 19

\[ \left \{ \ln \left ( x \right ) -2\,{\it \_C1}\,{\it Artanh} \left ( \left ( y \left ( x \right ) -2 \right ) {\it \_C1} \right ) -{\it \_C2}=0 \right \} \] Mathematica raw input

DSolve[x*y''[x] == (1 - y[x])*y'[x],y[x],x]

Mathematica raw output

{{y[x] -> 2 - Sqrt[2]*Sqrt[2 + C[1]]*Tanh[(Sqrt[2 + C[1]]*(2*C[2] - Log[x]))/Sqr
t[2]]}}

Maple raw input

dsolve(x*diff(diff(y(x),x),x) = (1-y(x))*diff(y(x),x), y(x),'implicit')

Maple raw output

ln(x)-2*_C1*arctanh((y(x)-2)*_C1)-_C2 = 0