Internal problem ID [11062]
Internal file name [OUTPUT/10319_Wednesday_January_24_2024_10_07_11_PM_18886836/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: 238.
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]]
\[ \boxed {x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y=0} \]
Writing the ode as \begin {align*} y^{\prime \prime } x^{2}-y^{\prime } x +\frac {a y}{x^{4}} = 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 &= 1\\ \beta &= \frac {\sqrt {a}}{2}\\ n &= {\frac {1}{2}}\\ \gamma &= -2 \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = \frac {2 c_{1} x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_{2} x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 c_{1} x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_{2} x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 c_{1} x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_{2} x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \] Verified OK.
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(x^6*diff(y(x),x$2)-x^5*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{2} \left (c_{1} \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_{2} \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.239 (sec). Leaf size: 58
DSolve[x^6*y''[x]-x^5*y'[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} x^2 e^{-\frac {i \sqrt {a}}{2 x^2}} \left (2 c_1 e^{\frac {i \sqrt {a}}{x^2}}-\frac {i c_2}{\sqrt {a}}\right ) \]