Internal problem ID [10966]
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: 132.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 85
dsolve(x^2*diff(y(x),x$2)+a*x*diff(y(x),x)+(b*x^n+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right )+c_{2} x^{-\frac {a}{2}+\frac {1}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.287 (sec). Leaf size: 168
DSolve[x^2*y''[x]+a*x*y'[x]+(b*x^n+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to n^{\frac {a-1}{n}} b^{-\frac {a-1}{2 n}} \left (x^n\right )^{-\frac {a-1}{2 n}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-2 a-4 c+1}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-2 a-4 c+1}}{n},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )+c_2 \operatorname {Gamma}\left (\frac {n+\sqrt {a^2-2 a-4 c+1}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-2 a-4 c+1}}{n},\frac {2 \sqrt {b} \sqrt {x^n}}{n}\right )\right ) \]