ODE
\[ y''(x)=a x \left (y'(x)^2+1\right )^{3/2} \] ODE Classification
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.740831 (sec), leaf count = 332
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {\frac {a x^2-2+2 c_1}{-1+c_1}} \sqrt {\frac {a x^2+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {a^2 x^4+4 a c_1 x^2-4+4 c_1{}^2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {a x^2-2+2 c_1}{-1+c_1}} \sqrt {\frac {a x^2+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {a^2 x^4+4 a c_1 x^2-4+4 c_1{}^2}}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.679 (sec), leaf count = 49
\[\left [y \left (x \right ) = \int \sqrt {-\frac {1}{a^{2} x^{4}+4 \textit {\_C1} \,a^{2} x^{2}+4 a^{2} \textit {\_C1}^{2}-4}}\, a \left (x^{2}+2 \textit {\_C1} \right )d x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[y''[x] == a*x*(1 + y'[x]^2)^(3/2),y[x],x]
Mathematica raw output
{{y[x] -> C[2] - (Sqrt[(-2 + a*x^2 + 2*C[1])/(-1 + C[1])]*Sqrt[(2 + a*x^2 + 2*C[1])/(1 + C[1])]*((-1 + C[1])*EllipticE[I*ArcSinh[x*Sqrt[a/(2 + 2*C[1])]], (1 + C[1])/(-1 + C[1])] + EllipticF[I*ArcSinh[x*Sqrt[a/(2 + 2*C[1])]], (1 + C[1])/(-1 + C[1])]))/(Sqrt[a/(2 + 2*C[1])]*Sqrt[-4 + a^2*x^4 + 4*a*x^2*C[1] + 4*C[1]^2])}, {y[x] -> C[2] + (Sqrt[(-2 + a*x^2 + 2*C[1])/(-1 + C[1])]*Sqrt[(2 + a*x^2 + 2*C[1])/(1 + C[1])]*((-1 + C[1])*EllipticE[I*ArcSinh[x*Sqrt[a/(2 + 2*C[1])]], (1 + C[1])/(-1 + C[1])] + EllipticF[I*ArcSinh[x*Sqrt[a/(2 + 2*C[1])]], (1 + C[1])/(-1 + C[1])]))/(Sqrt[a/(2 + 2*C[1])]*Sqrt[-4 + a^2*x^4 + 4*a*x^2*C[1] + 4*C[1]^2])}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*x*(1+diff(y(x),x)^2)^(3/2), y(x))
Maple raw output
[y(x) = Int((-1/(a^2*x^4+4*_C1*a^2*x^2+4*_C1^2*a^2-4))^(1/2)*a*(x^2+2*_C1),x)+_C2]