ODE
\[ y''(x)=a y(x) \left (y'(x)^2+1\right )^{3/2} \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.1069 (sec), leaf count = 350
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 1.424 (sec), leaf count = 106
\[\left [\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a}^{2}+2 \textit {\_C1} \right ) a}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_C1} \,\textit {\_a}^{2} a^{2}-4 a^{2} \textit {\_C1}^{2}+4}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {\left (\textit {\_a}^{2}+2 \textit {\_C1} \right ) a}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_C1} \,\textit {\_a}^{2} a^{2}-4 a^{2} \textit {\_C1}^{2}+4}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y''[x] == a*y[x]*(1 + y'[x]^2)^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((((-1 + C[1])*EllipticE[I*ArcSinh[Sqrt[a/(2 + 2*C[1])]*#1], (1 + C[1])/(-1 + C[1])] + EllipticF[I*ArcSinh[Sqrt[a/(2 + 2*C[1])]*#1], (1 + C[1])/(-1 + C[1])])*Sqrt[(-2 + 2*C[1] + a*#1^2)/(-1 + C[1])]*Sqrt[(2 + 2*C[1] + a*#1^2)/(1 + C[1])])/(Sqrt[a/(2 + 2*C[1])]*Sqrt[-4 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4])) & ][x + C[2]]}, {y[x] -> InverseFunction[(((-1 + C[1])*EllipticE[I*ArcSinh[Sqrt[a/(2 + 2*C[1])]*#1], (1 + C[1])/(-1 + C[1])] + EllipticF[I*ArcSinh[Sqrt[a/(2 + 2*C[1])]*#1], (1 + C[1])/(-1 + C[1])])*Sqrt[(-2 + 2*C[1] + a*#1^2)/(-1 + C[1])]*Sqrt[(2 + 2*C[1] + a*#1^2)/(1 + C[1])])/(Sqrt[a/(2 + 2*C[1])]*Sqrt[-4 + 4*C[1]^2 + 4*a*C[1]*#1^2 + a^2*#1^4]) & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*y(x)*(1+diff(y(x),x)^2)^(3/2), y(x))
Maple raw output
[Intat(1/(-_a^4*a^2-4*_C1*_a^2*a^2-4*_C1^2*a^2+4)^(1/2)*(_a^2+2*_C1)*a,_a = y(x))-x-_C2 = 0, Intat(-1/(-_a^4*a^2-4*_C1*_a^2*a^2-4*_C1^2*a^2+4)^(1/2)*(_a^2+2*_C1)*a,_a = y(x))-x-_C2 = 0]