61.28.18 problem 78

Internal problem ID [12578]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 78
Date solved : Tuesday, January 28, 2025 at 08:02:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.719 (sec). Leaf size: 102

dsolve(x*diff(y(x),x$2)-(2*a*x+1)*diff(y(x),x)+b*x^3*y(x)=0,y(x), singsol=all)
 

\[ y = x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{{1}/{4}}}, i \left (-b \right )^{{1}/{4}} x \right ) {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}} \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{-x^{2} \sqrt {-b}}}{\operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{{1}/{4}}}, i \left (-b \right )^{{1}/{4}} x \right )^{2} x^{3}}d x \right )\right ) \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[x*D[y[x],{x,2}]-(2*a*x+1)*D[y[x],x]+b*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved