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 = 0.342094 (sec), leaf count = 332
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {\frac {a x^2+2 c_1-2}{c_1-1}} \sqrt {\frac {a x^2+2 c_1+2}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2 c_1+2}} \sqrt {a^2 x^4+4 a c_1 x^2+4 c_1^2-4}}\right \},\left \{y(x)\to c_2+\frac {\sqrt {\frac {a x^2+2 c_1-2}{c_1-1}} \sqrt {\frac {a x^2+2 c_1+2}{c_1+1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )+\left (c_1-1\right ) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2 c_1+2}} \sqrt {a^2 x^4+4 a c_1 x^2+4 c_1^2-4}}\right \}\right \}\]
Maple ✓
cpu = 0.164 (sec), leaf count = 38
\[ \left \{ y \left ( x \right ) =\int \!\sqrt {- \left ( -4+ \left ( {x}^{2}+2\,{\it \_C1} \right ) ^{2}{a}^{2} \right ) ^{-1}}a \left ( {x}^{2}+2\,{\it \_C1} \right ) \,{\rm d}x+{\it \_C2} \right \} \] 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),'implicit')
Maple raw output
y(x) = Int((-1/(-4+(x^2+2*_C1)^2*a^2))^(1/2)*a*(x^2+2*_C1),x)+_C2