Internal problem ID [10959]
Internal file name [OUTPUT/10216_Sunday_December_31_2023_11_10_06_AM_94347684/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 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 136.
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.
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y=0} \]
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 -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunD ODE, case c = 0 <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.485 (sec). Leaf size: 243
dsolve(x^2*diff(y(x),x$2)+(a*x^2+b)*diff(y(x),x)+c*((a-c)*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {HeunD}\left (-4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right ) {\mathrm e}^{-x \left (a -c \right )} c_{2} +\operatorname {HeunD}\left (4 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}-1+\left (-4 a +8 c \right ) b , 8 \sqrt {-b \left (-2 c +a \right )}, -4 \sqrt {-b \left (-2 c +a \right )}+1+\left (-8 c +4 a \right ) b , \frac {\sqrt {-b \left (-2 c +a \right )}\, x -b}{\sqrt {-b \left (-2 c +a \right )}\, x +b}\right ) {\mathrm e}^{\frac {-c \,x^{2}+b}{x}} c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 1.026 (sec). Leaf size: 44
DSolve[x^2*y''[x]+(a*x^2+b)*y'[x]+c*((a-c)*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-c x} \left (c_2 \int _1^xe^{\frac {b}{K[1]}-a K[1]+2 c K[1]}dK[1]+c_1\right ) \]