4.18.37 \((3 x+5) y'(x)^2-(3 y(x)+3) y'(x)+y(x)=0\)

ODE
\[ (3 x+5) y'(x)^2-(3 y(x)+3) y'(x)+y(x)=0 \] ODE Classification

[_rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)

Mathematica
cpu = 2.59372 (sec), leaf count = 65

\[\text {Solve}\left [\left \{x=\frac {1}{3} e^{-3 \text {K$\$$1539465}} \left (3 c_1 (3 \text {K$\$$1539465}-1)+(9-27 \text {K$\$$1539465}) \text {Ei}(3 \text {K$\$$1539465})+4 e^{3 \text {K$\$$1539465}}\right ),y(x)=\frac {\text {K$\$$1539465} (\text {K$\$$1539465} (3 x+5)-3)}{3 \text {K$\$$1539465}-1}\right \},\{y(x),\text {K$\$$1539465}\}\right ]\]

Maple
cpu = 0.058 (sec), leaf count = 97

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac { \left (27\,{\it \_T}-9 \right ) {\it Ei} \left (1,-3\,{\it \_T} \right ) +4\,{{\rm e}^{3\,{\it \_T}}}+ \left (9\,{\it \_T}-3 \right ) {\it \_C1}}{3\, \left ({{\rm e}^{{\it \_T}}} \right ) ^{3}}},y \left ({\it \_T} \right ) ={\frac {1}{ \left ({{\rm e}^{{\it \_T}}} \right ) ^{3} \left (3\,{\it \_T}-1 \right ) } \left (27\,{{\it \_T}}^{2} \left ({\it \_T}-1/3 \right ) {\it Ei} \left (1,-3\,{\it \_T} \right ) +4\,{{\it \_T}}^{2}{{\rm e}^{3\,{\it \_T}}}+ \left (5\,{{\it \_T}}^{2}-3\,{\it \_T} \right ) \left ({{\rm e}^{{\it \_T}}} \right ) ^{3}+9\,{{\it \_T}}^{2}{\it \_C1}\, \left ({\it \_T}-1/3 \right ) \right ) }] \right \} \] Mathematica raw input

DSolve[y[x] - (3 + 3*y[x])*y'[x] + (5 + 3*x)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{x == (4*E^(3*K$1539465) + 3*(-1 + 3*K$1539465)*C[1] + (9 - 27*K$1539465)*
ExpIntegralEi[3*K$1539465])/(3*E^(3*K$1539465)), y[x] == (K$1539465*(-3 + K$1539
465*(5 + 3*x)))/(-1 + 3*K$1539465)}, {y[x], K$1539465}]

Maple raw input

dsolve((5+3*x)*diff(y(x),x)^2-(3+3*y(x))*diff(y(x),x)+y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/3*((27*_T-9)*Ei(1,-3*_T)+4*exp(3*_T)+(9*_T-3)*_C1)/exp(_T)^3, y(_T) =
 (27*_T^2*(_T-1/3)*Ei(1,-3*_T)+4*_T^2*exp(3*_T)+(5*_T^2-3*_T)*exp(_T)^3+9*_T^2*_
C1*(_T-1/3))/(3*_T-1)/exp(_T)^3]