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 = 0.156293 (sec), leaf count = 21
\[\left \{\left \{y(x)\to c_1 \left (x+\frac {1}{3}\right )-2+\frac {3}{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.042 (sec), leaf count = 49
\[\left [y \left (x \right ) = -2-2 \sqrt {1+3 x}, y \left (x \right ) = -2+2 \sqrt {1+3 x}, y \left (x \right ) = x \textit {\_C1} +\frac {\textit {\_C1}^{2}-6 \textit {\_C1} +9}{3 \textit {\_C1}}\right ]\] 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] -> -2 + 3/C[1] + (1/3 + x)*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))
Maple raw output
[y(x) = -2-2*(1+3*x)^(1/2), y(x) = -2+2*(1+3*x)^(1/2), y(x) = x*_C1+1/3*(_C1^2-6*_C1+9)/_C1]