Internal problem ID [9996]
Internal file name [OUTPUT/8943_Monday_June_06_2022_06_01_14_AM_89600235/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1675 (book 6.84).
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]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a^{2}+a b \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 &= a +\frac {b}{2}\\ \beta &= c\\ n &= -{\frac {1}{2}}\\ \gamma &= b \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = \frac {c_{1} x^{a +\frac {b}{2}} \sqrt {2}\, \cos \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}}+\frac {c_{2} x^{a +\frac {b}{2}} \sqrt {2}\, \sin \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} x^{a +\frac {b}{2}} \sqrt {2}\, \cos \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}}+\frac {c_{2} x^{a +\frac {b}{2}} \sqrt {2}\, \sin \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} x^{a +\frac {b}{2}} \sqrt {2}\, \cos \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}}+\frac {c_{2} x^{a +\frac {b}{2}} \sqrt {2}\, \sin \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}} \] Verified OK.
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 25
dsolve(x^2*diff(diff(y(x),x),x)-(2*a+b-1)*x*diff(y(x),x)+(c^2*b^2*x^(2*b)+a*(a+b))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{a} \left (c_{1} \sin \left (x^{b} c \right )+c_{2} \cos \left (x^{b} c \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.156 (sec). Leaf size: 69
DSolve[(a*(a + b) + b^2*c^2*x^(2*b))*y[x] - (-1 + 2*a + b)*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{-\frac {a+b}{b}} c^{a/b} \left (x^{2 b}\right )^{\frac {a}{2 b}} \left (2 c_1 \cos \left (c \sqrt {x^{2 b}}\right )+c_2 \sin \left (c \sqrt {x^{2 b}}\right )\right ) \]