4.34.16 \(a x^2 y'(x)+y(x) (b+c x)+x^3 y''(x)=0\)

ODE
\[ a x^2 y'(x)+y(x) (b+c x)+x^3 y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0658637 (sec), leaf count = 133

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

Maple ✓
cpu = 0.026 (sec), leaf count = 65

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

DSolve[(b + c*x)*y[x] + a*x^2*y'[x] + x^3*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> b^((-1 + a)/2)*(x^(-1))^((-1 + a)/2)*(BesselJ[-Sqrt[1 - 2*a + a^2 - 4*
c], 2*Sqrt[b]*Sqrt[x^(-1)]]*C[1]*Gamma[1 - Sqrt[1 - 2*a + a^2 - 4*c]] + BesselJ[
Sqrt[1 - 2*a + a^2 - 4*c], 2*Sqrt[b]*Sqrt[x^(-1)]]*C[2]*Gamma[1 + Sqrt[1 - 2*a +
 a^2 - 4*c]])}}

Maple raw input

dsolve(x^3*diff(diff(y(x),x),x)+a*x^2*diff(y(x),x)+(c*x+b)*y(x) = 0, y(x),'implicit')

Maple raw output

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