4.30.16 \(y(x) \left (a+b x^2\right )+x^2 y''(x)+2 x y'(x)=0\)

ODE
\[ y(x) \left (a+b x^2\right )+x^2 y''(x)+2 x y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0104063 (sec), leaf count = 58

\[\left \{\left \{y(x)\to c_1 j_{\frac {1}{2} \left (\sqrt {1-4 a}-1\right )}\left (\sqrt {b} x\right )+c_2 y_{\frac {1}{2} \left (\sqrt {1-4 a}-1\right )}\left (\sqrt {b} x\right )\right \}\right \}\]

Maple ✓
cpu = 0.06 (sec), leaf count = 43

\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\,{{\sl Y}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left (\sqrt {b}x\right )}+{\it \_C1}\,{{\sl J}_{{\frac {1}{2}\sqrt {1-4\,a}}}\left (\sqrt {b}x\right )} \right ) {\frac {1}{\sqrt {x}}}} \right \} \] Mathematica raw input

DSolve[(a + b*x^2)*y[x] + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*SphericalBesselJ[(-1 + Sqrt[1 - 4*a])/2, Sqrt[b]*x] + C[2]*Spheri
calBesselY[(-1 + Sqrt[1 - 4*a])/2, Sqrt[b]*x]}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+(b*x^2+a)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (_C2*BesselY(1/2*(1-4*a)^(1/2),b^(1/2)*x)+_C1*BesselJ(1/2*(1-4*a)^(1/2),b
^(1/2)*x))/x^(1/2)