ODE
\[ \left (1-x^2\right ) \left (y'''(x)\right )^2+2 x y''(x) y'''(x)-y''(x)^2+1=0 \] ODE Classification
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.211927 (sec), leaf count = 75
\[\left \{\left \{y(x)\to \frac {c_1 x^3}{6}-\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2\right \},\left \{y(x)\to \frac {c_1 x^3}{6}+\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.585 (sec), leaf count = 97
\[\left [y \left (x \right ) = -\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{6}+\frac {x \arcsin \left (x \right )}{2}+\frac {\sqrt {-x^{2}+1}}{2}+\textit {\_C1} x +\textit {\_C2}, y \left (x \right ) = \frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{6}-\frac {x \arcsin \left (x \right )}{2}-\frac {\sqrt {-x^{2}+1}}{2}+\textit {\_C1} x +\textit {\_C2}, y \left (x \right ) = \frac {\sqrt {\textit {\_C1}^{2}-1}\, x^{3}}{6}+\frac {x^{2} \textit {\_C1}}{2}+\textit {\_C2} x +\textit {\_C3}\right ]\] Mathematica raw input
DSolve[1 - y''[x]^2 + 2*x*y''[x]*y'''[x] + (1 - x^2)*y'''[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^3*C[1])/6 - (x^2*Sqrt[1 + C[1]^2])/2 + C[2] + x*C[3]}, {y[x] -> (x^3*C[1])/6 + (x^2*Sqrt[1 + C[1]^2])/2 + C[2] + x*C[3]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(diff(y(x),x),x),x)^2+2*x*diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)^2+1 = 0, y(x))
Maple raw output
[y(x) = -1/6*(-x^2+1)^(3/2)+1/2*x*arcsin(x)+1/2*(-x^2+1)^(1/2)+_C1*x+_C2, y(x) = 1/6*(-x^2+1)^(3/2)-1/2*x*arcsin(x)-1/2*(-x^2+1)^(1/2)+_C1*x+_C2, y(x) = 1/6*(_C1^2-1)^(1/2)*x^3+1/2*x^2*_C1+_C2*x+_C3]