problem 119

Internal problem ID [10942]
Internal ﬁle name [OUTPUT/10199_Sunday_December_31_2023_11_06_31_AM_12918249/index.tex]

Book: Handbook of exact solutions for ordinary diﬀerential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Diﬀerential Equations. section 2.1.2-4 Equation of form \(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 119.
ODE order: 2.
ODE degree: 1.

\[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y=0} \]

29.10.1 Solving as second order bessel ode ode

Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (-a^{2} x^{2 n}-2 a \,x^{n} b -a \,x^{n} n +a \,x^{n}-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 &= 2\\ n &= -2 b +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 b +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 b +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 b +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 b +1, 2 \sqrt {x}\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-2 b +1, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-2 b +1, 2 \sqrt {x}\right ) \] Veriﬁed OK.

Maple trace Kovacic algorithm successful

`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 
      A Liouvillian solution exists 
      Reducible group (found an exponential solution) 
      Group is reducible, not completely reducible 
   <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful`

\[ y \left (x \right ) = 2 \left (b -\frac {1}{2}-\frac {n}{2}\right )^{2} c_{2} x^{-\frac {3 n}{2}+\frac {1}{2}} \operatorname {WhittakerM}\left (\frac {n -2 b +1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+n \left (\left (-b +\frac {1}{2}+\frac {n}{2}\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}+x^{-\frac {n}{2}+\frac {1}{2}} a \right ) c_{2} \operatorname {WhittakerM}\left (-\frac {2 b +n -1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+c_{1} x^{b} {\mathrm e}^{\frac {a \,x^{n}}{n}} \]

29.10 problem 119

29.10.1 Solving as second order bessel ode ode