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 = 0.120954 (sec), leaf count = 95
\[\left \{\left \{y(x)\to c_3-\frac {i \sqrt {c_1^2 x^2+2 c_2 c_1^2 x+c_2^2 c_1^2-1}}{c_1}\right \},\left \{y(x)\to c_3+\frac {i \sqrt {c_1^2 x^2+2 c_2 c_1^2 x+c_2^2 c_1^2-1}}{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.222 (sec), leaf count = 49
\[ \left \{ y \left ( x \right ) =-\sqrt {-{{\it \_C2}}^{2}-2\,{\it \_C2}\,x-{x}^{2}+{\it \_C1}}+{\it \_C3},y \left ( x \right ) =\sqrt {-{{\it \_C2}}^{2}-2\,{\it \_C2}\,x-{x}^{2}+{\it \_C1}}+{\it \_C3} \right \} \] 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),'implicit')
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