4.18.36 $(3x+1)y^{\prime }(x{)}^2-3(y(x)+2)y^{\prime }(x)+9=0$

ODE
$$(3x+1)y^{\prime }(x{)}^2-3(y(x)+2)y^{\prime }(x)+9=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries], _Clairaut]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 0.386552 (sec), leaf count = 150

$$\{\{y(x)\to -\frac{\sqrt{(\sinh \left(2c_1\right)+\cosh \left(2c_1\right))((3x-34)\cosh \left(\frac{c_1}{2}\right)-3(x-12)\sinh \left(\frac{c_1}{2}\right)){}^2}+8\sinh \left(c_1\right)+8\cosh \left(c_1\right)-18x-294}{\sinh \left(c_1\right)+\cosh \left(c_1\right)-36}\},\{y(x)\to \frac{\sqrt{(\sinh \left(2c_1\right)+\cosh \left(2c_1\right))((3x-34)\cosh \left(\frac{c_1}{2}\right)-3(x-12)\sinh \left(\frac{c_1}{2}\right)){}^2}-8\sinh \left(c_1\right)-8\cosh \left(c_1\right)+18x+294}{\sinh \left(c_1\right)+\cosh \left(c_1\right)-36}\}\}$$

Maple ✓
cpu = 0.03 (sec), leaf count = 37

$$\{{\left(y\left(x\right)\right)}^2+4y\left(x\right)-12x=0,y\left(x\right)=\frac{9+(1+3x){\mathit{\_}\mathit{C}\mathit{1}}^2-6\mathit{\_}\mathit{C}\mathit{1}}{3\mathit{\_}\mathit{C}\mathit{1}}\}$$ 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 + Cos
h[C[1]] + Sinh[C[1]]))}, {y[x] -> (294 + 18*x - 8*Cosh[C[1]] - 8*Sinh[C[1]] + Sq
rt[((-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