Internal problem ID [6617]
Internal file name [OUTPUT/5865_Sunday_June_05_2022_03_58_53_PM_60445809/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS
ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 36 (a).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode", "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_Emden, _Fowler]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} y^{\prime \prime }+y=0} \] 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. \[ x^{3} y^{\prime \prime }+y = 0 \] 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 {1}{x^{3}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([\infty ]\)
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 <- Bessel successful <- special function solution successful`
✗ Solution by Maple
Order:=8; dsolve(x^3*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 294
AsymptoticDSolveValue[x^3*y''[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {11100458801337530625 i x^{13/2}}{4611686018427387904}+\frac {1327867167401775 i x^{11/2}}{4503599627370496}-\frac {468131288625 i x^{9/2}}{8796093022208}+\frac {66891825 i x^{7/2}}{4294967296}-\frac {72765 i x^{5/2}}{8388608}+\frac {105 i x^{3/2}}{8192}+\frac {1149690375852815671875 x^7}{147573952589676412928}-\frac {232376754295310625 x^6}{288230376151711744}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}-\frac {3 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {11100458801337530625 i x^{13/2}}{4611686018427387904}-\frac {1327867167401775 i x^{11/2}}{4503599627370496}+\frac {468131288625 i x^{9/2}}{8796093022208}-\frac {66891825 i x^{7/2}}{4294967296}+\frac {72765 i x^{5/2}}{8388608}-\frac {105 i x^{3/2}}{8192}+\frac {1149690375852815671875 x^7}{147573952589676412928}-\frac {232376754295310625 x^6}{288230376151711744}+\frac {33424574007825 x^5}{281474976710656}-\frac {14783093325 x^4}{549755813888}+\frac {2837835 x^3}{268435456}-\frac {4725 x^2}{524288}+\frac {15 x}{512}+\frac {3 i \sqrt {x}}{16}+1\right ) \]