#### 4.1834

ODE

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

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

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