ODE
\[ \left (y(x) y''(x)+y'(x)^2+1\right )^2=\left (y'(x)^2+1\right )^3 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 11.6813 (sec), leaf count = 270
\[\left \{\left \{y(x)\to -\sqrt {e^{-2 c_1} \left (1-e^{2 c_1} \left (c_2+x\right ){}^2\right )}-e^{-c_1}\right \},\left \{y(x)\to \sqrt {e^{-2 c_1} \left (1-e^{2 c_1} \left (c_2+x\right ){}^2\right )}-e^{-c_1}\right \},\left \{y(x)\to -\sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}-e^{c_1}\right \},\left \{y(x)\to e^{c_1}-\sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}-e^{c_1}\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}+e^{c_1}\right \},\left \{y(x)\to -e^{-c_1} \left (\sqrt {1-e^{2 c_1} \left (c_2+x\right ){}^2}-1\right )\right \},\left \{y(x)\to e^{-c_1} \left (\sqrt {1-e^{2 c_1} \left (c_2+x\right ){}^2}+1\right )\right \}\right \}\]
Maple ✓
cpu = 1.691 (sec), leaf count = 129
\[ \left \{ -{\it \_C2}+{y \left ( x \right ) \left ( 2\,{\it \_C1}-y \left ( x \right ) \right ) {\frac {1}{\sqrt {2\,{\it \_C1}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2}}}}}+x=0,-{\it \_C2}+{y \left ( x \right ) \left ( 2\,{\it \_C1}+y \left ( x \right ) \right ) {\frac {1}{\sqrt {-2\,{\it \_C1}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2}}}}}+x=0,{y \left ( x \right ) \left ( 2\,{\it \_C1}-y \left ( x \right ) \right ) {\frac {1}{\sqrt {2\,{\it \_C1}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2}}}}}-x-{\it \_C2}=0,{y \left ( x \right ) \left ( 2\,{\it \_C1}+y \left ( x \right ) \right ) {\frac {1}{\sqrt {-2\,{\it \_C1}\,y \left ( x \right ) - \left ( y \left ( x \right ) \right ) ^{2}}}}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[(1 + y'[x]^2 + y[x]*y''[x])^2 == (1 + y'[x]^2)^3,y[x],x]
Mathematica raw output
{{y[x] -> -E^(-C[1]) - Sqrt[(1 - E^(2*C[1])*(x + C[2])^2)/E^(2*C[1])]}, {y[x] -> -E^(-C[1]) + Sqrt[(1 - E^(2*C[1])*(x + C[2])^2)/E^(2*C[1])]}, {y[x] -> -E^C[1] - Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> E^C[1] - Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> -E^C[1] + Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> E^C[1] + Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> -((-1 + Sqrt[1 - E^(2*C[1])*(x + C[2])^2])/E^C[1])}, {y[x] -> (1 + Sqrt[1 - E^(2*C[1])*(x + C[2])^2])/E^C[1]}}
Maple raw input
dsolve((1+diff(y(x),x)^2+y(x)*diff(diff(y(x),x),x))^2 = (1+diff(y(x),x)^2)^3, y(x),'implicit')
Maple raw output
-_C2+y(x)*(2*_C1-y(x))/(2*_C1*y(x)-y(x)^2)^(1/2)+x = 0, -_C2+y(x)*(2*_C1+y(x))/(-2*_C1*y(x)-y(x)^2)^(1/2)+x = 0, y(x)*(2*_C1-y(x))/(2*_C1*y(x)-y(x)^2)^(1/2)-x-_C2 = 0, y(x)*(2*_C1+y(x))/(-2*_C1*y(x)-y(x)^2)^(1/2)-x-_C2 = 0