Internal problem ID [9518]
Internal file name [OUTPUT/8458_Monday_June_06_2022_03_07_25_AM_42075552/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1189.
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]]
\[ \boxed {x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{m}+c \right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{m}+c \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}-\frac {a}{2}\\ \beta &= \frac {2 \sqrt {b}}{m}\\ n &= \frac {\sqrt {a^{2}-2 a -4 c +1}}{m}\\ \gamma &= \frac {m}{2} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) \] 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: 0.047 (sec). Leaf size: 80
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^m+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {a}{2}} \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a -4 c +1}}{m}, \frac {2 \sqrt {b}\, x^{\frac {m}{2}}}{m}\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.193 (sec). Leaf size: 168
DSolve[(c + b*x^m)*y[x] + a*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to m^{\frac {a-1}{m}} b^{-\frac {a-1}{2 m}} \left (x^m\right )^{-\frac {a-1}{2 m}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-2 a-4 c+1}}{m}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-2 a-4 c+1}}{m},\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )+c_2 \operatorname {Gamma}\left (\frac {m+\sqrt {a^2-2 a-4 c+1}}{m}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-2 a-4 c+1}}{m},\frac {2 \sqrt {b} \sqrt {x^m}}{m}\right )\right ) \]