4.18.1 \(x y'(x)^2+y'(x)=y(x)\)

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

[_rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 1.23018 (sec), leaf count = 32

\[\text {Solve}\left [\left \{x=\frac {c_1-\text {K$\$$1057}+\log (\text {K$\$$1057})}{(\text {K$\$$1057}-1)^2},\text {K$\$$1057}^2 x+\text {K$\$$1057}=y(x)\right \},\{y(x),\text {K$\$$1057}\}\right ]\]

Maple
cpu = 0.018 (sec), leaf count = 39

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

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

Mathematica raw output

Solve[{x == (-K$1057 + C[1] + Log[K$1057])/(-1 + K$1057)^2, K$1057 + K$1057^2*x 
== y[x]}, {y[x], K$1057}]

Maple raw input

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

Maple raw output

[x(_T) = 1/(_T-1)^2*(-_T+ln(_T)+_C1), y(_T) = _T*(_T*ln(_T)+1+(_C1-2)*_T)/(_T-1)
^2]