Internal problem ID [9348]
Internal file name [OUTPUT/8284_Monday_June_06_2022_02_37_53_AM_94238721/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1015.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode"
Maple gives the following as the ode type
[_Titchmarsh]
\[ \boxed {y^{\prime \prime }-\left (a^{2} x^{2 n}-1\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (-x^{2} a^{2} x^{2 n}+x^{2}\right ) y = 0\tag {1} \end {align*}
Bessel ode has the form \begin {align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end {align*}
The generalized form of Bessel ode is given by Bowman (1958) as the following \begin {align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end {align*}
With the standard solution \begin {align*} y&=x^{\alpha } \left (c_{1} \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_{2} \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end {align*}
Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives \begin {align*} \alpha &= {\frac {1}{2}}\\ \beta &= \frac {2 \ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\\ n &= -\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\\ \gamma &= \frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-x^{2} a^{2} {\mathrm e}^{2 n \ln \left (x \right )}+x^{2}\right )}{2 \ln \left (x \right )}}}{\ln \left (-x^{2} \left ({\mathrm e}^{n \ln \left (x \right )} a -1\right ) \left ({\mathrm e}^{n \ln \left (x \right )} a +1\right )\right )}\right ) \] Verified OK.
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 an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> 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: 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 <- unable to find a useful change of variables trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)-(a^2*x^(2*n)-1)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(1 - a^2*x^(2*n))*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved