##### 4.25.15 $$a x^k y(x)+y''(x)=0$$

ODE
$a x^k y(x)+y''(x)=0$ ODE Classiﬁcation

[[_Emden, _Fowler]]

Book solution method
TO DO

Mathematica
cpu = 0.174038 (sec), leaf count = 112

$\left \{\left \{y(x)\to (k+2)^{-\frac {1}{k+2}} \sqrt {x} a^{\frac {1}{2 k+4}} \left (c_1 \Gamma \left (\frac {k+1}{k+2}\right ) J_{-\frac {1}{k+2}}\left (\frac {2 \sqrt {a} x^{\frac {k}{2}+1}}{k+2}\right )+c_2 \Gamma \left (1+\frac {1}{k+2}\right ) J_{\frac {1}{k+2}}\left (\frac {2 \sqrt {a} x^{\frac {k}{2}+1}}{k+2}\right )\right )\right \}\right \}$

Maple
cpu = 1.362 (sec), leaf count = 61

$\left [y \left (x \right ) = \textit {\_C1} \sqrt {x}\, \BesselJ \left (\frac {1}{k +2}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}+1}}{k +2}\right )+\textit {\_C2} \sqrt {x}\, \BesselY \left (\frac {1}{k +2}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}+1}}{k +2}\right )\right ]$ Mathematica raw input

DSolve[a*x^k*y[x] + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (a^(4 + 2*k)^(-1)*Sqrt[x]*(BesselJ[-(2 + k)^(-1), (2*Sqrt[a]*x^(1 + k/
2))/(2 + k)]*C[1]*Gamma[(1 + k)/(2 + k)] + BesselJ[(2 + k)^(-1), (2*Sqrt[a]*x^(1
 + k/2))/(2 + k)]*C[2]*Gamma[1 + (2 + k)^(-1)]))/(2 + k)^(2 + k)^(-1)}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*x^k*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*x^(1/2)*BesselJ(1/(k+2),2*a^(1/2)*x^(1/2*k+1)/(k+2))+_C2*x^(1/2)*Bes
selY(1/(k+2),2*a^(1/2)*x^(1/2*k+1)/(k+2))]