Internal problem ID [10857]
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 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 33.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.297 (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)
\[ y \left (x \right ) = c_{1} \operatorname {HeunT}\left (-\frac {a 3^{\frac {2}{3}} b}{\left (a^{2}\right )^{\frac {1}{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{\frac {x \left (3 b^{2}-x^{2}\right ) \operatorname {csgn}\left (a \right ) a \left (\operatorname {csgn}\left (a \right )+1\right )}{6}}+c_{2} \operatorname {HeunT}\left (-\frac {a 3^{\frac {2}{3}} b}{\left (a^{2}\right )^{\frac {1}{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{\frac {x \left (3 b^{2}-x^{2}\right ) \operatorname {csgn}\left (a \right ) a \left (\operatorname {csgn}\left (a \right )-1\right )}{6}} \]
✓ Solution by Mathematica
Time used: 3.893 (sec). Leaf size: 55
DSolve[y''[x]+a*(x^2-b^2)*y'[x]-a*(x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(b-x) \left (c_2 \int _1^x\frac {e^{a b^2 K[1]-\frac {1}{3} a K[1]^3}}{(b-K[1])^2}dK[1]+c_1\right )}{b} \]