Internal problem ID [11008]
Internal file name [OUTPUT/10265_Sunday_December_31_2023_11_33_32_AM_61588335/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 185.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 146
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-a} \left (-c x c_{2} \left (a b -c \right ) \left (b -c \right ) \operatorname {KummerU}\left (\frac {\left (a +1\right ) b -c}{b}, a , \frac {b}{x}\right )+\left (c_{1} x b \left (a b -c \right ) \operatorname {KummerM}\left (\frac {\left (a +1\right ) b -c}{b}, a , \frac {b}{x}\right )-\left (b c_{1} \operatorname {KummerM}\left (\frac {a b -c}{b}, a , \frac {b}{x}\right )-c c_{2} \operatorname {KummerU}\left (\frac {a b -c}{b}, a , \frac {b}{x}\right )\right ) \left (a b x +b^{2}-2 c x \right )\right ) b \right )}{b^{2} c} \]
✓ Solution by Mathematica
Time used: 0.435 (sec). Leaf size: 62
DSolve[x^3*y''[x]+(a*x^2+b*x)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric1F1}\left (-\frac {c}{b},2-a,\frac {b}{x}\right )-(-1)^a c_2 b^{a-1} \left (\frac {1}{x}\right )^{a-1} \operatorname {Hypergeometric1F1}\left (a-\frac {b+c}{b},a,\frac {b}{x}\right ) \]