Internal problem ID [11005]
Internal file name [OUTPUT/10262_Sunday_December_31_2023_11_33_28_AM_16378108/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 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 182.
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]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (a x +b \right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (a +\frac {b}{x}\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 \sqrt {b}\\ 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}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\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}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right ) \] 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 <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
dsolve(x^3*diff(y(x),x$2)+(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (-\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right ) c_{1} +\operatorname {BesselY}\left (-\sqrt {-4 a +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right ) c_{2} \right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.128 (sec). Leaf size: 101
DSolve[x^3*y''[x]+(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 \operatorname {Gamma}\left (1-\sqrt {1-4 a}\right ) \operatorname {BesselJ}\left (-\sqrt {1-4 a},2 \sqrt {b} \sqrt {\frac {1}{x}}\right )+c_2 \operatorname {Gamma}\left (\sqrt {1-4 a}+1\right ) \operatorname {BesselJ}\left (\sqrt {1-4 a},2 \sqrt {b} \sqrt {\frac {1}{x}}\right )}{\sqrt {b} \sqrt {\frac {1}{x}}} \]