Internal problem ID [6441]
Internal file name [OUTPUT/5689_Sunday_June_05_2022_03_47_04_PM_30345490/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR
SINGULAR POINTS. Page 175
Problem number: 1(a).
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.
\[ \boxed {x^{3} \left (x -1\right ) y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+3 x y=0} \] 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. \[ \left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (-2 x +2\right ) y^{\prime }+3 x y = 0 \] 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^{3}}\\ q(x) &= \frac {3}{x^{2} \left (x -1\right )}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([1, \infty ]\)
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 -> Kummer -> 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 differential order: 2; exact nonlinear 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:=8; dsolve(x^3*(x-1)*diff(y(x),x$2)-2*(x-1)*diff(y(x),x)+3*x*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 108
AsymptoticDSolveValue[x^3*(x-1)*y''[x]-2*(x-1)*y'[x]+3*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 e^{-\frac {1}{x^2}} \left (\frac {1731 x^7}{320}-\frac {795 x^6}{128}-\frac {51 x^5}{40}+\frac {63 x^4}{32}+\frac {x^3}{2}-\frac {3 x^2}{4}+1\right ) x^3+c_1 \left (-\frac {51 x^7}{320}-\frac {19 x^6}{128}-\frac {9 x^5}{40}-\frac {9 x^4}{32}-\frac {x^3}{2}-\frac {3 x^2}{4}+1\right ) \]