Internal problem ID [9549]
Internal file name [OUTPUT/8489_Monday_June_06_2022_03_12_58_AM_16411108/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1220.
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.
\[ \boxed {x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y=0} \]
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: 0.016 (sec). Leaf size: 38
dsolve(x^2*diff(diff(y(x),x),x)+2*x^2*f(x)*diff(y(x),x)+(x^2*(diff(f(x),x)+f(x)^2+a)-v*(v-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, {\mathrm e}^{-\left (\int f \left (x \right )d x \right )} \left (\operatorname {BesselJ}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{1} +\operatorname {BesselY}\left (v -\frac {1}{2}, x \sqrt {a}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.38 (sec). Leaf size: 62
DSolve[y[x]*((1 - v)*v + x^2*(a + f[x]^2 + Derivative[1][f][x])) + 2*x^2*f[x]*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (c_1 \operatorname {BesselJ}\left (v-\frac {1}{2},\sqrt {a} x\right )+c_2 \operatorname {BesselY}\left (v-\frac {1}{2},\sqrt {a} x\right )\right ) \exp \left (\int _1^x\left (\frac {1}{2 K[1]}-f(K[1])\right )dK[1]\right ) \]