Internal problem ID [9550]
Internal file name [OUTPUT/8490_Monday_June_06_2022_03_13_07_AM_46727516/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1221.
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.
\[ \boxed {x^{2} y^{\prime \prime }+\left (x -2 x^{2} f \left (x \right )\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y=0} \]
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 -> 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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(x^2*diff(diff(y(x),x),x)+(x-2*x^2*f(x))*diff(y(x),x)+(x^2*(1+f(x)^2-diff(f(x),x))-x*f(x)-v^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{\frac {\left (\int \frac {2 f \left (x \right ) x -1}{x}d x \right )}{2}} \left (c_{1} \operatorname {BesselJ}\left (v , x\right )+c_{2} \operatorname {BesselY}\left (v , x\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.206 (sec). Leaf size: 31
DSolve[y[x]*(-v^2 - x*f[x] + x^2*(1 + f[x]^2 - Derivative[1][f][x])) + (x - 2*x^2*f[x])*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x)) \exp \left (\int _1^xf(K[1])dK[1]\right ) \]