Internal problem ID [9695]
Internal file name [OUTPUT/8637_Monday_June_06_2022_04_34_02_AM_52787288/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1368.
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.
\[ \boxed {y^{\prime \prime }+\frac {a x y^{\prime }}{x^{2}+1}+\frac {b y}{\left (x^{2}+1\right )^{2}}=0} \]
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 <- Legendre successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 71
dsolve(diff(diff(y(x),x),x) = -a*x/(x^2+1)*diff(y(x),x)-b/(x^2+1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (\operatorname {LegendreP}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{1} +\operatorname {LegendreQ}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 92
DSolve[y''[x] == -((b*y[x])/(1 + x^2)^2) - (a*x*y'[x])/(1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)+c_2 Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)\right ) \]