ODE
\[ a y(x) \left (y'(x)^2+1\right )^2+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 10.947 (sec), leaf count = 262
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 (-a)+1+2 c_1}{1+2 c_1}} \sqrt {2 \text {$\#$1}^2 a-4 c_1} E\left (\sin ^{-1}\left (\sqrt {\frac {a}{2 c_1+1}} \text {$\#$1}\right )|1+\frac {1}{2 c_1}\right )}{\sqrt {\frac {a}{1+2 c_1}} \sqrt {\text {$\#$1}^2 (-a)+1+2 c_1} \sqrt {2-\frac {\text {$\#$1}^2 a}{c_1}}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 (-a)+1+2 c_1}{1+2 c_1}} \sqrt {2 \text {$\#$1}^2 a-4 c_1} E\left (\sin ^{-1}\left (\sqrt {\frac {a}{2 c_1+1}} \text {$\#$1}\right )|1+\frac {1}{2 c_1}\right )}{\sqrt {\frac {a}{1+2 c_1}} \sqrt {\text {$\#$1}^2 (-a)+1+2 c_1} \sqrt {2-\frac {\text {$\#$1}^2 a}{c_1}}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.61 (sec), leaf count = 94
\[\left [\int _{}^{y \left (x \right )}\frac {a \left (\textit {\_a}^{2}+2 \textit {\_C1} \right )}{\sqrt {-a \left (\textit {\_a}^{2}+2 \textit {\_C1} \right ) \left (a \,\textit {\_a}^{2}+2 \textit {\_C1} a -1\right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {a \left (\textit {\_a}^{2}+2 \textit {\_C1} \right )}{\sqrt {-a \left (\textit {\_a}^{2}+2 \textit {\_C1} \right ) \left (a \,\textit {\_a}^{2}+2 \textit {\_C1} a -1\right )}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[a*y[x]*(1 + y'[x]^2)^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((EllipticE[ArcSin[Sqrt[a/(1 + 2*C[1])]*#1], 1 + 1/(2*C[1])]*Sqrt[(1 + 2*C[1] - a*#1^2)/(1 + 2*C[1])]*Sqrt[-4*C[1] + 2*a*#1^2])/(Sqrt[a/(1 + 2*C[1])]*Sqrt[1 + 2*C[1] - a*#1^2]*Sqrt[2 - (a*#1^2)/C[1]])) & ][x + C[2]]}, {y[x] -> InverseFunction[(EllipticE[ArcSin[Sqrt[a/(1 + 2*C[1])]*#1], 1 + 1/(2*C[1])]*Sqrt[(1 + 2*C[1] - a*#1^2)/(1 + 2*C[1])]*Sqrt[-4*C[1] + 2*a*#1^2])/(Sqrt[a/(1 + 2*C[1])]*Sqrt[1 + 2*C[1] - a*#1^2]*Sqrt[2 - (a*#1^2)/C[1]]) & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*y(x)*(1+diff(y(x),x)^2)^2 = 0, y(x))
Maple raw output
[Intat(1/(-a*(_a^2+2*_C1)*(_a^2*a+2*_C1*a-1))^(1/2)*a*(_a^2+2*_C1),_a = y(x))-x-_C2 = 0, Intat(-1/(-a*(_a^2+2*_C1)*(_a^2*a+2*_C1*a-1))^(1/2)*a*(_a^2+2*_C1),_a = y(x))-x-_C2 = 0]