61.27.24 problem 34

Internal problem ID [12534]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 34
Date solved : Tuesday, January 28, 2025 at 03:20:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 11.407 (sec). Leaf size: 134

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

\[ y = c_{1} \operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+3 b \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}+3 b -6 c \right )}{6}}+c_{2} \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+3 b \right ) \operatorname {csgn}\left (a \right )-a \,x^{2}-3 b +6 c \right )}{6}} \]

✓ Solution by Mathematica

Time used: 1.118 (sec). Leaf size: 46

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

\[ y(x)\to e^{-c x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (a K[1]^2+3 b-6 c\right )}dK[1]+c_1\right ) \]