Internal problem ID [10952]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 118.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{n} a +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
dsolve(x^2*diff(y(x),x$2)+(a*x^n+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.343 (sec). Leaf size: 351
DSolve[x^2*y''[x]+(a*x^n+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to n^{-\frac {\sqrt {(1-4 b) n^2}+i \sqrt {4 b-1} n+n}{n^2}} a^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (x^n\right )^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (c_2 n^{\frac {2 \sqrt {(1-4 b) n^2}}{n^2}} a^{\frac {i \sqrt {4 b-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 b-1}}{n}} \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )+c_1 n^{\frac {2 i \sqrt {4 b-1}}{n}} a^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )\right ) \]