Internal problem ID [9670]
Internal file name [OUTPUT/8612_Monday_June_06_2022_04_27_59_AM_54443057/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1343.
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]]
\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (-a^{2}+a -\frac {b^{2}}{x^{2}}-\frac {b}{x}\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 &= 2\\ n &= -2 a +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 (-2 a +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 a +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 (-2 a +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 a +1, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-2 a +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 a +1, 2 \sqrt {x}\right ) \] 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 <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 58
dsolve(diff(diff(y(x),x),x) = -(x^2*a*(1-a)-b*(x+b))/x^4*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {BesselI}\left (a +1, \frac {b}{x}\right ) c_{1} b -\operatorname {BesselK}\left (a +1, \frac {b}{x}\right ) c_{2} b +2 \left (a x +\frac {b}{2}\right ) \left (\operatorname {BesselI}\left (a , \frac {b}{x}\right ) c_{1} +c_{2} \operatorname {BesselK}\left (a , \frac {b}{x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.275 (sec). Leaf size: 65
DSolve[y''[x] == -((((1 - a)*a*x^2 - b*(b + x))*y[x])/x^4),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 (2 a x+b) \operatorname {BesselI}\left (a,\frac {b}{x}\right )+b c_1 \operatorname {BesselI}\left (a+1,\frac {b}{x}\right )+c_2 \left ((2 a x+b) K_a\left (\frac {b}{x}\right )-b K_{a+1}\left (\frac {b}{x}\right )\right ) \]