4.28.22 \(a y'(x)+b y(x)+x y''(x)=0\)

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

[[_Emden, _Fowler]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.188497 (sec), leaf count = 77

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

Maple ✓
cpu = 0.029 (sec), leaf count = 47

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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