Internal problem ID [10833]
Internal file name [OUTPUT/9815_Sunday_June_19_2022_09_26_07_PM_32066164/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: 10.
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]]
\[ \boxed {y^{\prime \prime }+\left (x^{2 n} a +b \,x^{n -1}\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\left (x^{2} x^{2 n} a +b \,x^{n} x \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}}\\ \beta &= 2\\ n &= -1\\ \gamma &= {\frac {1}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] 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 -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 89
dsolve(diff(y(x),x$2)+(a*x^(2*n)+b*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {n}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x \,x^{n}}{n +1}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x \,x^{n}}{n +1}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.399 (sec). Leaf size: 225
DSolve[y''[x]+(a*x^(2*n)+b*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} e^{-\frac {\sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}},\frac {n}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_2 L_{\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right ) \]