Internal problem ID [10966]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 142.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 232
dsolve(x^2*diff(y(x),x$2)+x*(a*x^2+b*x+c)*diff(y(x),x)+(A*x^3+B*x^2+C*x+d)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {c}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-a^{2} x -2 a b +2 A \right )}{2 a}} \left (c_{1} x^{\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+c_{2} x^{-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*y''[x]+x*(a*x^2+b*x+c)*y'[x]+(A*x^3+B*x^2+C0*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved