Internal problem ID [10830]
Internal file name [OUTPUT/9812_Sunday_June_19_2022_09_26_03_PM_88912945/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 Equations Containing
Power Functions. page 213
Problem number: 7.
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]]
\[ \boxed {y^{\prime \prime }-x^{n} y a=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }-a \,x^{n} x^{2} 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 +2}\\ n &= -\frac {1}{n +2}\\ \gamma &= 1+\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 {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right ) \] 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: 0.281 (sec). Leaf size: 63
dsolve(diff(y(x),x$2)-a*x^n*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 119
DSolve[y''[x]-a*x^n*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (n+2)^{-\frac {1}{n+2}} \sqrt {x} a^{\frac {1}{2 n+4}} \left (c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselI}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )+c_2 (-1)^{\frac {1}{n+2}} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselI}\left (\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )\right ) \]