Internal problem ID [10917]
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 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 83.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (c -1\right ) \left (a x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 112
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)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \operatorname {HeunB}\left (c -1, \frac {b \sqrt {2}}{\sqrt {a}}, c -3, -\frac {\sqrt {2}\, b \left (c -2\right )}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+c_{2} x^{-c +1} {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \operatorname {HeunB}\left (-c +1, \frac {b \sqrt {2}}{\sqrt {a}}, c -3, -\frac {\sqrt {2}\, b \left (c -2\right )}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 1.579 (sec). Leaf size: 49
DSolve[x*y''[x]+(a*x^2+b*x+c)*y'[x]+(c-1)*(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{1-c} \left (c_2 \int _1^xe^{-\frac {1}{2} K[1] (2 b+a K[1])} K[1]^{c-2}dK[1]+c_1\right ) \]