28.7 problem 67

Internal problem ID [10890]
Internal file name [OUTPUT/10147_Sunday_December_24_2023_05_15_10_PM_64440882/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 2.1.2-3 Equation of form \((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 67.
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

[[_Emden, _Fowler]]

\[ \boxed {x y^{\prime \prime }+a y^{\prime }+b \,x^{n} y=0} \]

28.7.1 Solving as second order bessel ode ode

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

Verification of solutions

\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (-\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (-\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) \] 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: 0.218 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.245 (sec). Leaf size: 165

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

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