4.1834    ′′       ( ′  2   )3∕2
y (x) = ax y(x) + 1

ODE

 ′′       ( ′  2   )3∕2
y (x) = ax y(x) + 1

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

(|(|            ∘ ax2+2c1−2∘ ax2+2c1+2(  (     −1( ∘ --a--) c1+1)           (    − 1( ∘ --a-)  c1+1)) )|  (|           ∘ ax2+2c1−2∘ ax2+2c1+2-( (     −1(  ∘--a--) c1+1)           (    − 1&
{{            ----c1−1------c1+1----F--isinh---x---2c1+2-|c1−1-+-(c1 −-1)E-isinh----x--2c1+2--|c1−1---}  {           ----c1−1------c1+1----F--isinh---x---2c1+2-|c1−1-+-(c1 −-1)E-isinh----x--2c1+2--|c1−1---}}
|(|(y (x) → c2 −                           ∘ --a--∘a2x4-+-4acx2-+4c2-− 4                           |) ,|(y(x) → c2 +                           ∘ --a--∘a2x4-+-4acx2-+-4c2-−-4                           |)|)
                                           2c1+2           1      1                                                                           2c1+2           1      1

Maple
cpu = 0.164 (sec), leaf count = 38

{       ∫ ∘ ------------------------                   }
              (       2       2 2)−1 ( 2       )
  y(x) =    −  − 4+ (x + 2-C1) a    a x  +2 C1  dx+ -C2

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