61.27.23 problem 33

Internal problem ID [12533]
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 : 33
Date solved : Tuesday, January 28, 2025 at 03:20:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.979 (sec). Leaf size: 141

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

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

✓ Solution by Mathematica

Time used: 0.403 (sec). Leaf size: 60

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

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