61.31.1 problem 182

Internal problem ID [12682]
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
Problem number : 182
Date solved : Tuesday, January 28, 2025 at 03:38:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+\left (a x +b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 49

dsolve(x^3*diff(y(x),x$2)+(a*x+b)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 101

DSolve[x^3*D[y[x],{x,2}]+(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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