4.28.26 \(a y'(x)+b x^k y(x)+x y''(x)=0\)

ODE
\[ a y'(x)+b x^k y(x)+x y''(x)=0 \] ODE Classification

[[_Emden, _Fowler]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0554993 (sec), leaf count = 165

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

Maple ✓
cpu = 0.103 (sec), leaf count = 71

\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {a}{2}}+{\frac {1}{2}}} \left ( {{\sl Y}_{{\frac {a-1}{k+1}}}\left (2\,{\frac {\sqrt {b}{x}^{1/2+k/2}}{k+1}}\right )}{\it \_C2}+{{\sl J}_{{\frac {a-1}{k+1}}}\left (2\,{\frac {\sqrt {b}{x}^{1/2+k/2}}{k+1}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x) = x^(-1/2*a+1/2)*(BesselY((a-1)/(k+1),2*b^(1/2)*x^(1/2+1/2*k)/(k+1))*_C2+Be
sselJ((a-1)/(k+1),2*b^(1/2)*x^(1/2+1/2*k)/(k+1))*_C1)