ODE
\[ \left (y(x)^2+1\right ) y''(x)=3 y(x) y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0904146 (sec), leaf count = 93
\[\left \{\left \{y(x)\to -\frac {i c_1 \left (c_2+x\right )}{\sqrt {c_1^2 x^2+2 c_2 c_1^2 x+c_2^2 c_1^2-1}}\right \},\left \{y(x)\to \frac {i c_1 \left (c_2+x\right )}{\sqrt {c_1^2 x^2+2 c_2 c_1^2 x+c_2^2 c_1^2-1}}\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 22
\[ \left \{ {y \left ( x \right ) {\frac {1}{\sqrt {1+ \left ( y \left ( x \right ) \right ) ^{2}}}}}-{\it \_C1}\,x-{\it \_C2}=0 \right \} \] 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)*C[1]*(x + C[2]))/Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2]}, {y[x] -> (I*C[1]*(x + C[2]))/Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2]}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x) = 3*y(x)*diff(y(x),x)^2, y(x),'implicit')
Maple raw output
1/(1+y(x)^2)^(1/2)*y(x)-_C1*x-_C2 = 0