Internal problem ID [2920]
Internal file name [OUTPUT/2412_Sunday_June_05_2022_03_06_58_AM_35030435/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page
758
Problem number: 5.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )}=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )} = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= \frac {2}{x \left (x -3\right )}\\ q(x) &= -\frac {1}{x^{3} \left (x +3\right )}\\ \end {align*}
| \(p(x)=\frac {2}{x \left (x -3\right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 3\) | \(\text {``regular''}\) |
| \(q(x)=-\frac {1}{x^{3} \left (x +3\right )}\) | |
| singularity | type |
| \(x = -3\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``irregular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([3, -3, \infty ]\)
Irregular singular points : \([0]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
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 -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
Order:=6; dsolve(diff(y(x),x$2)+2/(x*(x-3))*diff(y(x),x)-1/(x^3*(x+3))*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.223 (sec). Leaf size: 258
AsymptoticDSolveValue[y''[x]+2/(x*(x-3))*y'[x]-1/(x^3*(x+3))*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 e^{-\frac {2}{\sqrt {3} \sqrt {x}}} \left (\frac {10879996003390494539 x^{9/2}}{6059672463464202240 \sqrt {3}}+\frac {64713480610417 x^{7/2}}{328758271672320 \sqrt {3}}+\frac {287821451 x^{5/2}}{3397386240 \sqrt {3}}+\frac {19817 x^{3/2}}{73728 \sqrt {3}}-\frac {4894564486149401320457 x^5}{1246561192484064460800}-\frac {116612812982297797 x^4}{378729528966512640}-\frac {22160647459 x^3}{587068342272}+\frac {463507 x^2}{42467328}+\frac {587 x}{4608}+\frac {25 \sqrt {x}}{16 \sqrt {3}}+1\right ) x^{13/12}+c_2 e^{\frac {2}{\sqrt {3} \sqrt {x}}} \left (-\frac {10879996003390494539 x^{9/2}}{6059672463464202240 \sqrt {3}}-\frac {64713480610417 x^{7/2}}{328758271672320 \sqrt {3}}-\frac {287821451 x^{5/2}}{3397386240 \sqrt {3}}-\frac {19817 x^{3/2}}{73728 \sqrt {3}}-\frac {4894564486149401320457 x^5}{1246561192484064460800}-\frac {116612812982297797 x^4}{378729528966512640}-\frac {22160647459 x^3}{587068342272}+\frac {463507 x^2}{42467328}+\frac {587 x}{4608}-\frac {25 \sqrt {x}}{16 \sqrt {3}}+1\right ) x^{13/12} \]