Internal problem ID [9543]
Internal file name [OUTPUT/8483_Monday_June_06_2022_03_11_48_AM_67876875/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1214.
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
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (\left (-1\right )^{n} a -x^{4}+\left (2 a +2 n +1\right ) x^{2}-a^{2}\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (-x^{4}+2 a \,x^{2}+2 n \,x^{2}+\left (-1\right )^{n} a -a^{2}+x^{2}\right ) y = 0\tag {1} \end {align*}
Bessel ode has the form \begin {align*} x^{2} y^{\prime \prime }+x y^{\prime }+\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 &= 2\\ n &= \sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}\\ \gamma &= {\frac {1}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 \left (-1\right )^{n} a +4 a^{2}+1}, 2 \sqrt {x}\right ) \] Verified OK.
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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 71
dsolve(x^2*diff(diff(y(x),x),x)+(-x^4+(2*n+2*a+1)*x^2+(-1)^n*a-a^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {WhittakerM}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{1} +\operatorname {WhittakerW}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.409 (sec). Leaf size: 191
DSolve[((-1)^n*a - a^2 + (1 + 2*a + 2*n)*x^2 - x^4)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )+c_2 L_{\frac {1}{4} \left (2 a+2 n-\sqrt {4 a^2-4 (-1)^n a+1}-1\right )}^{\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1}}\left (x^2\right )\right )}{\sqrt {x}} \]