Internal problem ID [9675]
Internal file name [OUTPUT/8617_Monday_June_06_2022_04_28_44_AM_12928444/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1348.
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]]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2} a +b +\frac {a}{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 &= 0\\ \beta &= 2\\ n &= 2 \sqrt {-b}\\ \gamma &= {\frac {1}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \operatorname {BesselJ}\left (2 \sqrt {-b}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {-b}, 2 \sqrt {x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \operatorname {BesselJ}\left (2 \sqrt {-b}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {-b}, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \operatorname {BesselJ}\left (2 \sqrt {-b}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {-b}, 2 \sqrt {x}\right ) \] 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 -> 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 trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunD ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.828 (sec). Leaf size: 73
dsolve(diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-(b*x^2+a*(x^4+1))/x^4*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right ) \left (c_{1} +c_{2} \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.523 (sec). Leaf size: 34
DSolve[y''[x] == -(((b*x^2 + a*(1 + x^4))*y[x])/x^4) - y'[x]/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \text {MathieuC}[-b,a,i \log (x)]+c_2 \text {MathieuS}[-b,a,i \log (x)] \]