Internal problem ID [5588]
Internal file name [OUTPUT/4836_Sunday_June_05_2022_03_07_31_PM_28516887/index.tex]
Book: A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications.
Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section: Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page
239
Problem number: 33.
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.
\[ \boxed {x^{4} y^{\prime \prime }+\lambda y=0} \] 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. \[ x^{4} y^{\prime \prime }+\lambda y = 0 \] 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 {\lambda }{x^{4}}\\ \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 Kovacic algorithm successful
`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 A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful`
✗ Solution by Maple
Order:=6; dsolve(x^4*diff(y(x),x$2)+lambda*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.104 (sec). Leaf size: 50
AsymptoticDSolveValue[x^4*y''[x]+\[Lambda]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x e^{\frac {i \sqrt {\lambda }}{x}}-\frac {i c_2 x e^{-\frac {i \sqrt {\lambda }}{x}}}{2 \sqrt {\lambda }} \]