29.9 problem 118

Internal problem ID [10941]
Internal file name [OUTPUT/10198_Sunday_December_31_2023_11_06_30_AM_50299776/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: 118.
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]]

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

29.9.1 Solving as second order bessel ode ode

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

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

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) \\ \end{align*}

Verification of solutions

\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) \] 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: 63

dsolve(x^2*diff(y(x),x$2)+(a*x^n+b)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.343 (sec). Leaf size: 351

DSolve[x^2*y''[x]+(a*x^n+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

\[ y(x)\to n^{-\frac {\sqrt {(1-4 b) n^2}+i \sqrt {4 b-1} n+n}{n^2}} a^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (x^n\right )^{\frac {-\sqrt {(1-4 b) n^2}-i \sqrt {4 b-1} n+n}{2 n^2}} \left (c_2 n^{\frac {2 \sqrt {(1-4 b) n^2}}{n^2}} a^{\frac {i \sqrt {4 b-1}}{n}} \left (x^n\right )^{\frac {i \sqrt {4 b-1}}{n}} \operatorname {Gamma}\left (\frac {n+\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )+c_1 n^{\frac {2 i \sqrt {4 b-1}}{n}} a^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \left (x^n\right )^{\frac {\sqrt {(1-4 b) n^2}}{n^2}} \operatorname {Gamma}\left (1-\frac {\sqrt {1-4 b}}{n}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {(1-4 b) n^2}}{n^2},\frac {2 \sqrt {a} \sqrt {x^n}}{n}\right )\right ) \]