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 = 0.195785 (sec), leaf count = 129
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [e^{2 c_1} \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {e^{2 c_1}-\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {e^{2 c_1}-\text {$\#$1}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}} \sqrt {e^{2 c_1}-\text {$\#$1}}-e^{2 c_1} \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {e^{2 c_1}-\text {$\#$1}}}\right )\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.108 (sec), leaf count = 87
\[ \left \{ -\sqrt {y \left ( x \right ) \left ( {\it \_C1}-y \left ( x \right ) \right ) }+{\frac {{\it \_C1}}{2}\arctan \left ( {1 \left ( y \left ( x \right ) -{\frac {{\it \_C1}}{2}} \right ) {\frac {1}{\sqrt {y \left ( x \right ) \left ( {\it \_C1}-y \left ( x \right ) \right ) }}}} \right ) }-x-{\it \_C2}=0,\sqrt {y \left ( x \right ) \left ( {\it \_C1}-y \left ( x \right ) \right ) }-{\frac {{\it \_C1}}{2}\arctan \left ( {1 \left ( y \left ( x \right ) -{\frac {{\it \_C1}}{2}} \right ) {\frac {1}{\sqrt {y \left ( x \right ) \left ( {\it \_C1}-y \left ( x \right ) \right ) }}}} \right ) }-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[1 + y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[E^(2*C[1])*ArcTan[Sqrt[#1]/Sqrt[E^(2*C[1]) - #1]] - Sqrt[E^(2*C[1]) - #1]*Sqrt[#1] & ][x + C[2]]}, {y[x] -> InverseFunction[-(E^(2*C[1])*ArcTan[Sqrt[#1]/Sqrt[E^(2*C[1]) - #1]]) + Sqrt[E^(2*C[1]) - #1]*Sqrt[#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),'implicit')
Maple raw output
-(y(x)*(_C1-y(x)))^(1/2)+1/2*_C1*arctan((y(x)-1/2*_C1)/(y(x)*(_C1-y(x)))^(1/2))-x-_C2 = 0, (y(x)*(_C1-y(x)))^(1/2)-1/2*_C1*arctan((y(x)-1/2*_C1)/(y(x)*(_C1-y(x)))^(1/2))-x-_C2 = 0