Internal
problem
ID
[8919]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
26
Date
solved
:
Wednesday, March 05, 2025 at 07:08:13 AM
CAS
classification
:
[[_2nd_order, _exact, _linear, _homogeneous]]
Solve
Using series expansion around
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE.
The following is summary of singularities for the above ode. Writing the ode as
Where
Combining everything together gives the following summary of singularities for the ode as
Regular singular points :
Irregular singular points :
Since
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
Substituting all the above in Eq (2A) gives the following equation where now all powers of
The indicial equation is obtained from
When
Or
Since
Since the above is true for all
Solving for
Since
Solving for
Since
Or
We start by finding
Solving for
Which for the root
At this point, it is a good idea to keep track of
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Which for the root
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Which for the root
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For
Which for the root
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For
Which for the root
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For
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Using the above table, then the solution
Now the second solution
Solving for
Which for the root
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Using the above table, then the solution
Therefore the homogeneous solution is
Hence the final solution is
Order:=6; ode:=x^2*(x+3)*diff(diff(y(x),x),x)+5*x*(x+1)*diff(y(x),x)-(1-4*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)] One independent solution has integrals. Trying a hypergeometric solution free of integrals... -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 2F1 ODE -> Trying to convert hypergeometric functions to elementary form... <- elementary form for at least one hypergeometric solution is achieved - returning with no uncomputed integrals <- linear_1 successful`
Maple step by step
ode=x^2*(3+x)*D[y[x],{x,2}] + 5*x*(1+x)*D[y[x],x] - (1-4*x)*y[x] == 0; ic={}; AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
from sympy import * x = symbols("x") y = Function("y") ode = Eq(x**2*(x + 3)*Derivative(y(x), (x, 2)) + 5*x*(x + 1)*Derivative(y(x), x) - (1 - 4*x)*y(x),0) ics = {} dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)