Internal problem ID [11006]
Internal file name [OUTPUT/10263_Sunday_December_31_2023_11_33_29_AM_57178898/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: 183.
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.
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y=0} \]
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: 0.156 (sec). Leaf size: 120
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b)*diff(y(x),x)+c*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \left (\operatorname {KummerM}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{1} +\operatorname {KummerU}\left (-\frac {1}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}+\frac {a}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.646 (sec). Leaf size: 308
DSolve[x^3*y''[x]+(a*x^2+b)*y'[x]+c*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -(-1)^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a+3\right )} 2^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}-a+1\right )} b^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 c+1}+a-1\right )} \left (c_2 i^{\sqrt {a^2-2 a-4 c+1}} b^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \left (\frac {1}{x}\right )^{\sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{2} \left (\sqrt {a^2-2 a-4 c+1}+2\right ),\frac {b}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {a^2-2 a-4 c+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),1-\frac {1}{2} \sqrt {a^2-2 a-4 c+1},\frac {b}{2 x^2}\right )\right ) \]