Internal
problem
ID
[8958]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
66
Date
solved
:
Wednesday, March 05, 2025 at 07:09:40 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
Solve
Using series expansion around
Solving ode using Taylor series method. This gives review on how the Taylor series method works for solving second order ode.
Let
Assuming expansion is at
But
And so on. Hence if we name
Therefore (6) can be used from now on along with
To find
And so on. Evaluating all the above at initial conditions
Substituting all the above in (7) and simplifying gives the solution as
Since the expansion point
Then
Substituting the above back into the ode gives
Which simplifies to
The next step is to make all powers of
Substituting all the above in Eq (2) gives the following equation where now all powers of
For
Solving for
For
Which after substituting the earlier terms found becomes
For
Which after substituting the earlier terms found becomes
For
Which after substituting the earlier terms found becomes
For
Which after substituting the earlier terms found becomes
For
Which after substituting the earlier terms found becomes
And so on. Therefore the solution is
Substituting the values for
Collecting terms, the solution becomes
At
Order:=6; ode:=diff(diff(y(x),x),x)+(-1+x)*y(x) = 0; dsolve(ode,y(x),type='series',x=0);
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`
Maple step by step
ode=D[y[x],{x,2}]+(x-1)*y[x]==0; ic={}; AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
from sympy import * x = symbols("x") y = Function("y") ode = Eq((x - 1)*y(x) + Derivative(y(x), (x, 2)),0) ics = {} dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)