Internal problem ID [9601]
Internal file name [OUTPUT/8542_Monday_June_06_2022_03_48_46_AM_89432881/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1273.
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]]
\[ \boxed {4 y^{\prime \prime } x^{2}-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y=0} \]
Writing the ode as \begin {align*} y^{\prime \prime } x^{2}+\left (k x -m^{2}-\frac {1}{4} x^{2}+\frac {1}{4}\right ) y = 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 &= {\frac {1}{2}}\\ \beta &= 2\\ n &= 2 m\\ \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 (2 m , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (2 m , 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 (2 m , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (2 m , 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (2 m , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (2 m , 2 \sqrt {x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 \left (\frac {d}{d x}y^{\prime }\right ) x^{2}+\left (4 k x -4 m^{2}-x^{2}+1\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {\left (4 k x -4 m^{2}-x^{2}+1\right ) y}{4 x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (4 k x -4 m^{2}-x^{2}+1\right ) y}{4 x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=0, P_{3}\left (x \right )=\frac {4 k x -4 m^{2}-x^{2}+1}{4 x^{2}}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-m^{2}+\frac {1}{4} \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 4 \left (\frac {d}{d x}y^{\prime }\right ) x^{2}+\left (4 k x -4 m^{2}-x^{2}+1\right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (2 r -1+2 m \right ) \left (2 r -1-2 m \right ) x^{r}+\left (a_{1} \left (2 r +1+2 m \right ) \left (2 r +1-2 m \right )+4 a_{0} k \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (2 r -1+2 m +2 k \right ) \left (2 r -1-2 m +2 k \right )+4 a_{k -1} k -a_{k -2}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (2 r -1+2 m \right ) \left (2 r -1-2 m \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {1}{2}-m , m +\frac {1}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (2 r +1+2 m \right ) \left (2 r +1-2 m \right )+4 a_{0} k =0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=\frac {4 a_{0} k}{4 m^{2}-4 r^{2}-4 r -1} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 4 \left (r -\frac {1}{2}-m +k \right ) \left (r -\frac {1}{2}+m +k \right ) a_{k}+4 a_{k -1} k -a_{k -2}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & 4 \left (r +\frac {3}{2}-m +k \right ) \left (r +\frac {3}{2}+m +k \right ) a_{k +2}+4 a_{k +1} k -a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {4 a_{k +1} k -a_{k}}{\left (2 r +3-2 m +2 k \right ) \left (2 r +3+2 m +2 k \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2}-m \\ {} & {} & a_{k +2}=-\frac {4 a_{k +1} k -a_{k}}{\left (4-4 m +2 k \right ) \left (4+2 k \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2}-m \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}-m}, a_{k +2}=-\frac {4 a_{k +1} k -a_{k}}{\left (4-4 m +2 k \right ) \left (4+2 k \right )}, a_{1}=\frac {4 a_{0} k}{4 m^{2}-4 \left (\frac {1}{2}-m \right )^{2}-3+4 m}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =m +\frac {1}{2} \\ {} & {} & a_{k +2}=-\frac {4 a_{k +1} k -a_{k}}{\left (4+2 k \right ) \left (4 m +4+2 k \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =m +\frac {1}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +m +\frac {1}{2}}, a_{k +2}=-\frac {4 a_{k +1} k -a_{k}}{\left (4+2 k \right ) \left (4 m +4+2 k \right )}, a_{1}=\frac {4 a_{0} k}{4 m^{2}-4 \left (m +\frac {1}{2}\right )^{2}-4 m -3}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +\frac {1}{2}-m}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +m +\frac {1}{2}}\right ), a_{n +2}=-\frac {4 k a_{n +1}-a_{n}}{\left (4-4 m +2 n \right ) \left (4+2 n \right )}, a_{1}=\frac {4 a_{0} k}{4 m^{2}-4 \left (\frac {1}{2}-m \right )^{2}-3+4 m}, b_{n +2}=-\frac {4 k b_{n +1}-b_{n}}{\left (4+2 n \right ) \left (4 m +4+2 n \right )}, b_{1}=\frac {4 b_{0} k}{4 m^{2}-4 \left (m +\frac {1}{2}\right )^{2}-4 m -3}\right ] \end {array} \]
Maple trace
`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 <- 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: 0.046 (sec). Leaf size: 17
dsolve(4*x^2*diff(diff(y(x),x),x)-(-4*k*x+4*m^2+x^2-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (k , m , x\right )+c_{2} \operatorname {WhittakerW}\left (k , m , x\right ) \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 20
DSolve[(1 - 4*m^2 + 4*k*x - x^2)*y[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 M_{k,m}(x)+c_2 W_{k,m}(x) \]