problem 133

Internal problem ID [10956]
Internal ﬁle name [OUTPUT/10213_Sunday_December_31_2023_11_09_46_AM_10467483/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: 133.
ODE order: 2.
ODE degree: 1.

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

29.24.1 Solving as second order bessel ode ode

Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2 n}+c \,x^{n}\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}-\frac {a}{2}\\ \beta &= 2\\ n &= -a +1\\ \gamma &= {\frac {1}{2}} \end {align*}

Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (-a +1, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (-a +1, 2 \sqrt {x}\right ) \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (-a +1, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (-a +1, 2 \sqrt {x}\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (-a +1, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (-a +1, 2 \sqrt {x}\right ) \] 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 
   -> 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`

\[ y \left (x \right ) = \left (\operatorname {WhittakerM}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right ) c_{2} \right ) x^{-\frac {a}{2}-\frac {n}{2}+\frac {1}{2}} \]

\[ y(x)\to 2^{\frac {a+n-1}{2 n}} x^{\frac {1}{2} (-a-n+1)} \left (x^n\right )^{\frac {a+n-1}{2 n}} e^{\frac {i \sqrt {b} x^n}{n}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {-a+\frac {i c}{\sqrt {b}}-n+1}{2 n},\frac {a+n-1}{n},-\frac {2 i \sqrt {b} x^n}{n}\right )+c_2 L_{-\frac {a-\frac {i c}{\sqrt {b}}+n-1}{2 n}}^{\frac {a-1}{n}}\left (-\frac {2 i \sqrt {b} x^n}{n}\right )\right ) \]

29.24 problem 133

29.24.1 Solving as second order bessel ode ode