ODE
\[ (3 x+1) y'(x)^2-3 (y(x)+2) y'(x)+9=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.386552 (sec), leaf count = 150
\[\left \{\left \{y(x)\to -\frac {\sqrt {\left (\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )\right ) \left ((3 x-34) \cosh \left (\frac {c_1}{2}\right )-3 (x-12) \sinh \left (\frac {c_1}{2}\right )\right ){}^2}+8 \sinh \left (c_1\right )+8 \cosh \left (c_1\right )-18 x-294}{\sinh \left (c_1\right )+\cosh \left (c_1\right )-36}\right \},\left \{y(x)\to \frac {\sqrt {\left (\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )\right ) \left ((3 x-34) \cosh \left (\frac {c_1}{2}\right )-3 (x-12) \sinh \left (\frac {c_1}{2}\right )\right ){}^2}-8 \sinh \left (c_1\right )-8 \cosh \left (c_1\right )+18 x+294}{\sinh \left (c_1\right )+\cosh \left (c_1\right )-36}\right \}\right \}\]
Maple ✓
cpu = 0.03 (sec), leaf count = 37
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+4\,y \left ( x \right ) -12\,x=0,y \left ( x \right ) ={\frac {9+ \left ( 1+3\,x \right ) {{\it \_C1}}^{2}-6\,{\it \_C1}}{3\,{\it \_C1}}} \right \} \] Mathematica raw input
DSolve[9 - 3*(2 + y[x])*y'[x] + (1 + 3*x)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((-294 - 18*x + 8*Cosh[C[1]] + 8*Sinh[C[1]] + Sqrt[((-34 + 3*x)*Cosh[C[1]/2] - 3*(-12 + x)*Sinh[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]])])/(-36 + Cosh[C[1]] + Sinh[C[1]]))}, {y[x] -> (294 + 18*x - 8*Cosh[C[1]] - 8*Sinh[C[1]] + Sqrt[((-34 + 3*x)*Cosh[C[1]/2] - 3*(-12 + x)*Sinh[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]])])/(-36 + Cosh[C[1]] + Sinh[C[1]])}}
Maple raw input
dsolve((1+3*x)*diff(y(x),x)^2-3*(2+y(x))*diff(y(x),x)+9 = 0, y(x),'implicit')
Maple raw output
y(x)^2+4*y(x)-12*x = 0, y(x) = 1/3*(9+(1+3*x)*_C1^2-6*_C1)/_C1