32.2 problem 212

Internal problem ID [11035]
Internal file name [OUTPUT/10292_Wednesday_January_24_2024_10_06_23_PM_69199443/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 2.1.2-7 Equation of form \((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 212.
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]]

\[ \boxed {x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y=0} \]

32.2.1 Solving as second order bessel ode ode

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

Verification of solutions

\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 a +1}, 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 a +1}, 2 \sqrt {x}\right ) \] Verified OK.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> 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`
 

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 63

dsolve(x^4*diff(y(x),x$2)+(a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = x \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {-4 a +1}}{2}, \frac {2 i \sqrt {c}}{x}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {c}}, \frac {\sqrt {-4 a +1}}{2}, \frac {2 i \sqrt {c}}{x}\right )\right ) \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[x^4*y''[x]+(a*x^2+b*x+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

Not solved