Internal
problem
ID
[8969]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
5.0
Problem
number
:
8
Date
solved
:
Wednesday, March 05, 2025 at 07:13:25 AM
CAS
classification
:
[[_linear, `class A`]]
Solve
Using series expansion around
Since this is an inhomogeneous, then let the solution be
Where
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 (2A) gives the following equation where now all powers of
The indicial equation is obtained from
When
The corresponding balance equation is found by replacing
This equation will used later to find the particular solution.
Since
Since the above is true for all
Solving for
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
Which can be written as
Collecting terms, the solution becomes
Finally, since
Now we determine the particular solution
For
The particular solution is therefore
Where in the above
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Unable to find particular solution .Unable to find the particular solution. No solution exist.
Order:=6; ode:=diff(y(x),x)+y(x) = 1/x^2; dsolve(ode,y(x),type='series',x=0);
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
Maple step by step
ode=D[y[x],x]+y[x]==1/x^2; ic={}; AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
from sympy import * x = symbols("x") y = Function("y") ode = Eq(y(x) + Derivative(y(x), x) - 1/x**2,0) ics = {} dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
ValueError : ODE y(x) + Derivative(y(x), x) - 1/x**2 does not match hint 1st_power_series