ODE
\[ a x^k y(x)+y''(x)=0 \] ODE Classification
[[_Emden, _Fowler]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0699298 (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 = 0.141 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) =\sqrt {x} \left ( {{\sl Y}_{ \left ( k+2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}{x}^{k/2+1}}{k+2}}\right )}{\it \_C2}+{{\sl J}_{ \left ( k+2 \right ) ^{-1}}\left (2\,{\frac {\sqrt {a}{x}^{k/2+1}}{k+2}}\right )}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = x^(1/2)*(BesselY(1/(k+2),2*a^(1/2)*x^(1/2*k+1)/(k+2))*_C2+BesselJ(1/(k+2),2*a^(1/2)*x^(1/2*k+1)/(k+2))*_C1)