ODE
\[ \left (y'(x)^2+1\right ) y'''(x)=3 y'(x) y''(x)^2 \] ODE Classification
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.382301 (sec), leaf count = 95
\[\left \{\left \{y(x)\to c_3-\frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}\right \},\left \{y(x)\to \frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}+c_3\right \}\right \}\]
Maple ✓
cpu = 1.921 (sec), leaf count = 49
\[\left [y \left (x \right ) = -\sqrt {-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}+\textit {\_C1}}+\textit {\_C3}, y \left (x \right ) = \sqrt {-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}+\textit {\_C1}}+\textit {\_C3}\right ]\] Mathematica raw input
DSolve[(1 + y'[x]^2)*y'''[x] == 3*y'[x]*y''[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2])/C[1] + C[3]}, {y[x] -> (I*Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2])/C[1] + C[3]}}
Maple raw input
dsolve((1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x) = 3*diff(y(x),x)*diff(diff(y(x),x),x)^2, y(x))
Maple raw output
[y(x) = -(-_C2^2-2*_C2*x-x^2+_C1)^(1/2)+_C3, y(x) = (-_C2^2-2*_C2*x-x^2+_C1)^(1/2)+_C3]