61.29.9 problem 118

Internal problem ID [12618]
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
Problem number : 118
Date solved : Tuesday, January 28, 2025 at 03:23:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 63

dsolve(x^2*diff(y(x),x$2)+(a*x^n+b)*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\operatorname {BesselJ}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.208 (sec). Leaf size: 351

DSolve[x^2*D[y[x],{x,2}]+(a*x^n+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to n^{-\frac {\sqrt {(1-4 b) n^2}+i \sqrt {4 b-1} n+n}{n^2}} a^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (x^n\right )^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (c_2 n^{\frac {2 \sqrt {(1-4 b) n^2}}{n^2}} a^{\frac {i \sqrt {4 b-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 b-1}}{n}} \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )+c_1 n^{\frac {2 i \sqrt {4 b-1}}{n}} a^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )\right ) \]