Internal problem ID [9428]
Internal file name [OUTPUT/8368_Monday_June_06_2022_02_51_28_AM_55196046/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1099.
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.
\[ \boxed {x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler 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 <- Bessel successful <- special function solution successful Change of variables used: [x = ln(t)^(1/2)] Linear ODE actually solved: (-v^2+t)*u(t)+4*t*diff(u(t),t)+4*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 25
dsolve(x*diff(y(x),x$2)-diff(y(x),x)+x^3*(exp(x^2)-v^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {x^{2}}{2}}\right )+c_{2} \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {x^{2}}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 1.118 (sec). Leaf size: 46
DSolve[x*y''[x]-y'[x]+x^3*(Exp[x^2]-v^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Gamma}(1-v) \operatorname {BesselJ}\left (-v,\sqrt {e^{x^2}}\right )+c_2 \operatorname {Gamma}(v+1) \operatorname {BesselJ}\left (v,\sqrt {e^{x^2}}\right ) \]