ODE
\[ 2 y(x) y''(x)+y'(x)^2+1=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.596113 (sec), leaf count = 166
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )+e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \sin ^{-1}\left (\sqrt {\text {$\#$1}} e^{-c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )+e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \sin ^{-1}\left (\sqrt {\text {$\#$1}} e^{-c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 2.562 (sec), leaf count = 823
\[\left [y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} \textit {\_C2} -4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} \textit {\_C2} -4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right ) \textit {\_C1} +2 \textit {\_C2} +2 x \right )}{2}+\frac {\textit {\_C1}}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} \textit {\_C2} -4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}-4 \textit {\_C1} \textit {\_Z} \textit {\_C2} -4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right ) \textit {\_C1} -2 \textit {\_C2} -2 x \right )}{2}+\frac {\textit {\_C1}}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} \textit {\_C2} +4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} \textit {\_C2} +4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right ) \textit {\_C1} -2 \textit {\_C2} -2 x \right )}{2}+\frac {\textit {\_C1}}{2}, y \left (x \right ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} \textit {\_C2} +4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1}^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} \textit {\_C2} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C1} x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_C2}^{2}+8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x \textit {\_C2} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}+4 \textit {\_C1} \textit {\_Z} \textit {\_C2} +4 \textit {\_C1} \textit {\_Z} x -\textit {\_C1}^{2}+4 \textit {\_C2}^{2}+8 \textit {\_C2} x +4 x^{2}\right ) \textit {\_C1} +2 \textit {\_C2} +2 x \right )}{2}+\frac {\textit {\_C1}}{2}\right ]\] Mathematica raw input
DSolve[1 + y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((Sqrt[#1]*(-E^(2*C[1]) + #1) + E^(3*C[1])*ArcSin[Sqrt[#1]/E^C[1]]*Sqrt[1 - #1/E^(2*C[1])])/Sqrt[E^(2*C[1]) - #1]) & ][x + C[2]]}, {y[x] -> InverseFunction[(Sqrt[#1]*(-E^(2*C[1]) + #1) + E^(3*C[1])*ArcSin[Sqrt[#1]/E^C[1]]*Sqrt[1 - #1/E^(2*C[1])])/Sqrt[E^(2*C[1]) - #1] & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1 = 0, y(x))
Maple raw output
[y(x) = 1/2*tan(RootOf(tan(_Z)^2*_C1^2*_Z^2-4*tan(_Z)^2*_C1*_C2*_Z-4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2-4*_C1*_Z*_C2-4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2))*(-RootOf(tan(_Z)^2*_C1^2*_Z^2-4*tan(_Z)^2*_C1*_C2*_Z-4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2-4*_C1*_Z*_C2-4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2)*_C1+2*_C2+2*x)+1/2*_C1, y(x) = 1/2*tan(RootOf(tan(_Z)^2*_C1^2*_Z^2-4*tan(_Z)^2*_C1*_C2*_Z-4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2-4*_C1*_Z*_C2-4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2))*(RootOf(tan(_Z)^2*_C1^2*_Z^2-4*tan(_Z)^2*_C1*_C2*_Z-4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2-4*_C1*_Z*_C2-4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2)*_C1-2*_C2-2*x)+1/2*_C1, y(x) = 1/2*tan(RootOf(tan(_Z)^2*_C1^2*_Z^2+4*tan(_Z)^2*_C1*_C2*_Z+4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2+4*_C1*_Z*_C2+4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2))*(-RootOf(tan(_Z)^2*_C1^2*_Z^2+4*tan(_Z)^2*_C1*_C2*_Z+4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2+4*_C1*_Z*_C2+4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2)*_C1-2*_C2-2*x)+1/2*_C1, y(x) = 1/2*tan(RootOf(tan(_Z)^2*_C1^2*_Z^2+4*tan(_Z)^2*_C1*_C2*_Z+4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2+4*_C1*_Z*_C2+4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2))*(RootOf(tan(_Z)^2*_C1^2*_Z^2+4*tan(_Z)^2*_C1*_C2*_Z+4*tan(_Z)^2*_C1*x*_Z+4*tan(_Z)^2*_C2^2+8*tan(_Z)^2*x*_C2+4*tan(_Z)^2*x^2+_C1^2*_Z^2+4*_C1*_Z*_C2+4*_C1*_Z*x-_C1^2+4*_C2^2+8*_C2*x+4*x^2)*_C1+2*_C2+2*x)+1/2*_C1]