problem 121

Internal problem ID [10944]
Internal ﬁle name [OUTPUT/10201_Sunday_December_31_2023_11_07_54_AM_35512270/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: 121.
ODE order: 2.
ODE degree: 1.

\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y=0} \]

29.12.1 Solving as second order bessel ode ode

Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\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 &= n\\ \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 (n , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n , 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 (n , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n , 2 \sqrt {x}\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (n , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n , 2 \sqrt {x}\right ) \] 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 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      <- hyper3 successful: indirect Equivalence to 0F1 under \`\`^ @ Moebius\`\` is resolved 
   <- hypergeometric successful 
<- special function solution successful`

\[ y \left (x \right ) = \frac {\left (\frac {2 \,3^{\frac {5}{6}} \pi c_{2} \left (a \,x^{n}+b \right ) \operatorname {BesselI}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right )}{3}+c_{1} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}} {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{3}}\right ) x^{-\frac {n}{2}+\frac {1}{2}}}{3 {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )} \]

29.12 problem 121

29.12.1 Solving as second order bessel ode ode