Internal problem ID [10902]
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: 78.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+y b \,x^{3}=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 106
dsolve(x*diff(y(x),x$2)-(2*a*x+1)*diff(y(x),x)+b*x^3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{2} \operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right ) {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}} \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{-x^{2} \sqrt {-b}}}{\operatorname {HeunB}\left (2, 0, \frac {a^{2}}{\sqrt {-b}}, -\frac {2 i a}{\left (-b \right )^{\frac {1}{4}}}, i \left (-b \right )^{\frac {1}{4}} x \right )^{2} x^{3}}d x \right )\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y''[x]-(2*a*x+1)*y'[x]+b*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved