Internal
problem
ID
[7951]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
10
Date
solved
:
Monday, October 21, 2024 at 04:39:24 PM
CAS
classification
:
[_quadrature]
Solve
Using series expansion around \(x=0\)
Since this is an inhomogeneous, then let the solution be
Where \(y_h\) is the solution to the homogeneous ode \(y^{\prime } = 0\),and \(y_p\) is a particular solution to the inhomogeneous ode. First, we solve for \(y_h\) Let the solution be represented as Frobenius power series of the form
Then
Substituting the above back into the ode gives
Which simplifies to
The next step is to make all powers of \(x\) be \(n +r -1\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r -1}\) and adjusting the power and the corresponding index gives Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r -1\).
The indicial equation is obtained from \(n=0\). From Eq (2) this gives
When \(n=0\) the above becomes
The corresponding balance equation is found by replacing \(r\) by \(m\) and \(a\) by \(c\) to avoid confusing terms between particular solution and the homogeneous solution. Hence the balance equation is
This equation will used later to find the particular solution.
Since \(a_{0}\neq 0\) then the indicial equation becomes
Since the above is true for all \(x\) then the indicial equation simplifies to
Solving for \(r\) gives the root of the indicial equation as
From the above we see that there is no recurrence relation since there is only one summation term. Therefore all \(a_{n}\) terms are zero except for \(a_{0}\). Hence
Unable to solve the balance equation \(m c_{0} x^{-1+m} = \frac {1}{x}\) for \(c_{0}\) and \(x\). No particular solution exists.
Unable to find the particular solution. No solution exist.
Methods for first order ODEs:
Solving time : 0.020
(sec)
Leaf size : maple_leaf_size
dsolve(diff(y(x),x) = 1/x,y(x), series,x=0)