Internal problem ID [9484]
Internal file name [OUTPUT/8424_Monday_June_06_2022_03_02_16_AM_73992342/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1155.
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]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{k}-b \left (b -1\right )\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (a \,x^{k}-b^{2}+b \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 \sqrt {a}}{k}\\ n &= -\frac {2 b -1}{k}\\ \gamma &= \frac {k}{2} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) \] 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 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: 0.047 (sec). Leaf size: 71
dsolve(x^2*diff(diff(y(x),x),x)+(a*x^k-b*(b-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.098 (sec). Leaf size: 116
DSolve[((1 - b)*b + a*x^k)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to k^{-1/k} a^{\left .\frac {1}{2}\right /k} \left (x^k\right )^{\left .\frac {1}{2}\right /k} \left (c_1 \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_2 \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right ) \]