Internal problem ID [11078]
Internal file name [OUTPUT/10335_Wednesday_January_24_2024_10_17_11_PM_46271332/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 2.1.2-8. Other equations.
Problem number: 255.
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]]
\[ \boxed {\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y=0} \]
Writing the ode as \begin {align*} y^{\prime \prime } x^{2}+\left (-\frac {x^{2 n} b^{2}}{x^{2}}-\frac {x^{n} a b n}{x^{2}}+\frac {b \,x^{2 n}}{x^{2}}+\frac {x^{n} a b}{x^{2}}\right ) y = 0\tag {1} \end {align*}
Bessel ode has the form \begin {align*} y^{\prime \prime } x^{2}+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*} y^{\prime \prime } x^{2}+\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 \ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\\ n &= -\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\\ \gamma &= \frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {\ln \left (x \right )}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}, \frac {2 \ln \left (x \right ) x^{\frac {\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}{2 \ln \left (x \right )}}}{\ln \left (-\frac {b \left ({\mathrm e}^{n \ln \left (x \right )} b +a n -{\mathrm e}^{n \ln \left (x \right )}-a \right )}{x^{2}}\right )+n \ln \left (x \right )}\right ) \] Verified OK.
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form could result into a too large expression - returning special function form of solution, free of uncomput <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 75
dsolve((x^n+a)^2*diff(y(x),x$2)-b*x^(n-2)*( (b-1)*x^n+a*(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{2} \left (a x +x^{n +1}\right ) \operatorname {hypergeom}\left (\left [1, \frac {n -2 b +1}{n}\right ], \left [1+\frac {1}{n}\right ], -\frac {x^{n}}{a}\right )+\left (\frac {x^{n}+a}{a}\right )^{\frac {2 b}{n}} a c_{1} \right ) \left (x^{n}+a \right )^{-\frac {b}{n}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(x^n+a)^2*y''[x]-b*x^(n-2)*( (b-1)*x^n+a*(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved