Internal problem ID [11007]
Internal file name [OUTPUT/10264_Sunday_December_31_2023_11_33_31_AM_3795090/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: 184.
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.
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+y b=0} \]
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Kummer successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form is not straightforward to achieve - returning special function solution free of uncomputed integrals <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 108
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) \left (-1\right )^{-a} \left (a -2\right ) b^{-a +1}+c_{1} \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) \left (-1\right )^{-a} b^{-a +2}+b \,x^{-a +1} c_{1} {\mathrm e}^{\frac {b}{x}}+c_{2} \left (a -2\right ) x -c_{1} x^{-a +2} {\mathrm e}^{\frac {b}{x}}+c_{2} b}{x} \]
✓ Solution by Mathematica
Time used: 2.653 (sec). Leaf size: 62
DSolve[x^3*y''[x]+(a*x^2+b*x)*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {((a-2) x+b) \left (c_2 \int _1^x\frac {e^{\frac {b}{K[1]}} K[1]^{2-a}}{(b+(a-2) K[1])^2}dK[1]+c_1\right )}{x (a+b-2)} \]