61.31.10 problem 191

Internal problem ID [12691]
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 : 191
Date solved : Tuesday, January 28, 2025 at 08:11:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.574 (sec). Leaf size: 169

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

\[ y = \frac {\left (a x +b \right )^{\frac {3 a -c}{a}} \left (c_{1} x^{\frac {-3 a +c}{a}} \operatorname {HeunC}\left (\frac {\lambda a}{b}, -\frac {c}{a}+1, 3-\frac {c}{a}, 0, -\frac {\lambda a}{b}+\frac {c \lambda }{2 b}+\frac {5}{2}-\frac {2 c}{a}+\frac {c^{2}}{2 a^{2}}, -\frac {b}{a x}\right ) x^{2}+c_{2} \operatorname {HeunC}\left (\frac {\lambda a}{b}, -1+\frac {c}{a}, 3-\frac {c}{a}, 0, -\frac {\lambda a}{b}+\frac {c \lambda }{2 b}+\frac {5}{2}-\frac {2 c}{a}+\frac {c^{2}}{2 a^{2}}, -\frac {b}{a x}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.739 (sec). Leaf size: 70

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

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