Internal problem ID [11016]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 182.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (a x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(x^3*diff(y(x),x$2)+(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\sqrt {1-4 a}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\sqrt {1-4 a}, \frac {2 \sqrt {b}}{\sqrt {x}}\right ) \]
✓ Solution by Mathematica
Time used: 0.128 (sec). Leaf size: 101
DSolve[x^3*y''[x]+(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 \operatorname {Gamma}\left (1-\sqrt {1-4 a}\right ) \operatorname {BesselJ}\left (-\sqrt {1-4 a},2 \sqrt {b} \sqrt {\frac {1}{x}}\right )+c_2 \operatorname {Gamma}\left (\sqrt {1-4 a}+1\right ) \operatorname {BesselJ}\left (\sqrt {1-4 a},2 \sqrt {b} \sqrt {\frac {1}{x}}\right )}{\sqrt {b} \sqrt {\frac {1}{x}}} \]