6.2 problem 2

Internal problem ID [4732]
Internal file name [OUTPUT/4225_Sunday_June_05_2022_12_43_44_PM_18743969/index.tex]

Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section: Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number: 2.
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]]

\[ \boxed {u^{\prime \prime }-\frac {a^{2} u}{x^{\frac {2}{3}}}=0} \]

6.2.1 Solving as second order bessel ode ode

Writing the ode as \begin {align*} x^{2} u^{\prime \prime }-a^{2} x^{\frac {4}{3}} u = 0\tag {1} \end {align*}

Bessel ode has the form \begin {align*} x^{2} u^{\prime \prime }+u^{\prime } x +\left (-n^{2}+x^{2}\right ) u = 0\tag {2} \end {align*}

The generalized form of Bessel ode is given by Bowman (1958) as the following \begin {align*} x^{2} u^{\prime \prime }+\left (1-2 \alpha \right ) x u^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) u = 0\tag {3} \end {align*}

With the standard solution \begin {align*} u&=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 {3 i a}{2}\\ n &= {\frac {3}{4}}\\ \gamma &= {\frac {2}{3}} \end {align*}

Substituting all the above into (4) gives the solution as \begin {align*} u = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right ) \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} u &= c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right ) \\ \end{align*}

Verification of solutions

\[ u = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right ) \] 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: 0.0 (sec). Leaf size: 41

dsolve(diff(u(x),x$2)-a^2*x^(-2/3)*u(x)=0,u(x), singsol=all)
 

\[ u \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right ) c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.05 (sec). Leaf size: 79

DSolve[u''[x]-a^2*x^(-2/3)*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

\[ u(x)\to \frac {3^{3/4} a^{3/4} \sqrt {x} \left (16 c_1 \operatorname {Gamma}\left (\frac {5}{4}\right ) \operatorname {BesselI}\left (-\frac {3}{4},\frac {3}{2} a x^{2/3}\right )+3 (-1)^{3/4} c_2 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselI}\left (\frac {3}{4},\frac {3}{2} a x^{2/3}\right )\right )}{8 \sqrt {2}} \]