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

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0299254 (sec), leaf count = 85

\[\left \{\left \{y(x)\to e^{-x/2} \sqrt {x} \left (c_1 J_{\frac {1}{2} \sqrt {1-4 a}}\left (-\frac {1}{2} i \sqrt {1-4 b} x\right )+c_2 Y_{\frac {1}{2} \sqrt {1-4 a}}\left (-\frac {1}{2} i \sqrt {1-4 b} x\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.068 (sec), leaf count = 57

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

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

Mathematica raw output

{{y[x] -> (Sqrt[x]*(BesselJ[Sqrt[1 - 4*a]/2, (-I/2)*Sqrt[1 - 4*b]*x]*C[1] + Bess
elY[Sqrt[1 - 4*a]/2, (-I/2)*Sqrt[1 - 4*b]*x]*C[2]))/E^(x/2)}}

Maple raw input

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

Maple raw output

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