Internal problem ID [6449]
Internal file name [OUTPUT/5697_Sunday_June_05_2022_03_47_33_PM_29724232/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: 2(e).
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.
\[ \boxed {x^{4} y^{\prime \prime }+\sin \left (x \right ) y=0} \] 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. \[ x^{4} y^{\prime \prime }+\sin \left (x \right ) y = 0 \] 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) &= 0\\ q(x) &= \frac {\sin \left (x \right )}{x^{4}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([]\)
Irregular singular points : \([0, \infty ]\)
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 symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> 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 changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> 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 with_periodic_functions in the coefficients --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 5 trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> 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 <- unable to find a useful change of variables 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 to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> 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 with_periodic_functions in the coefficients --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 5 --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
Order:=8; dsolve(x^4*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.11 (sec). Leaf size: 294
AsymptoticDSolveValue[x^4*y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}-\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}+\frac {12579783586699513 i x^{9/2}}{96185277197844480}-\frac {21896783401 i x^{7/2}}{579820584960}+\frac {856783 i x^{5/2}}{41943040}-\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {16487484152477478659746223 i x^{13/2}}{2773583263632691770163200}+\frac {4594934148364735183693 i x^{11/2}}{6320013947079701299200}-\frac {12579783586699513 i x^{9/2}}{96185277197844480}+\frac {21896783401 i x^{7/2}}{579820584960}-\frac {856783 i x^{5/2}}{41943040}+\frac {3151 i x^{3/2}}{73728}-\frac {3986263268940827572255963529 x^7}{207094217017907652172185600}+\frac {21730712888356628741772337 x^6}{10920984100553723845017600}-\frac {1500040357444099007 x^5}{5129881450551705600}+\frac {4885269094757 x^4}{74217034874880}-\frac {2835642457 x^3}{108716359680}+\frac {11659 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \]