61.28.23 problem 83

Internal problem ID [12583]
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 : 83
Date solved : Tuesday, January 28, 2025 at 03:22:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.109 (sec). Leaf size: 102

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

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

✓ Solution by Mathematica

Time used: 0.525 (sec). Leaf size: 57

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

\[ y(x)\to x^{1-c} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {a K[1]^2+b K[1]-c+2}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]