Question

Solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution.

y′′ + xy = 0, x0 = 0

Answer #1

Series Solution Method. Solve the given differential equation by
means of a power series about the given point x0. Find the
recurrence relation; also find the first four terms in each of two
linearly independent solutions (unless the series terminates
sooner). If possible, find the general term in each solution.
(1 − x)y′′ + y = 0, x0 = 0

Series Solutions of Ordinary Differential Equations For the
following problems solve the given differential equation by means
of a power series about the given point x0. Find the recurrence
relation; also find the first four terms in each of two linearly
independed sollutions (unless the series terminates sooner). If
possible, find the general term in each solution.
y"+k2x2y=0, x0=0,
k-constant

Use a power series centered about the ordinary point x0 = 0 to
solve the differential equation
(x − 4)y′′ − y′ + 12xy = 0
Find the recurrence relation and at least the first four nonzero
terms of each of the two linearly inde-
pendent solutions (unless the series terminates sooner).
What is the guaranteed radius of
convergence?

solve y'-y=0 about the point X0=0 by means of a power series.
Find the recurrence relation and two linearly independent
solutions. ( X0 meaning X naught)

Find the first four nonzero terms in a power series expansion
about x0 for a general solution to the given
differential equation with the given value for x0.
x2y''-y'+y = 0; x0 = 2

Use a series centered at x0=0 to find the general solution of
y"+x^2y'-2y=0. Use a series centered at x0=0 to find the general
solution. Write out at least 4 nonzero terms of each series
corresponding to the two linearly independent solutions.

Solve by using power series: y' =
x^5(y). Find the recurrence relation and compute the first 25
coefficients. Check your solution to the differential equation with
the original equation if possible, please.

Consider the differential equation
4x2y′′ − 8x2y′ + (4x2 + 1)y = 0
(a) Verify that x0 = 0 is a regular singular point of the
differential equation and then find one solution as a Frobenius
series centered at x0 = 0. The indicial equation has a single root
with multiplicity two. Therefore the differential equation has only
one Frobenius series solution. Write your solution in terms of
familiar elementary functions.
(b) Use Reduction of Order to find a second...

Find a power series solution of the given differential equation.
Write the solution in terms of power series of familiar elementary
functions.
a. (3? − 1)?′ + 3? = 0
b. ?′ − 10?? = 0

Solve the following differential equation by assuming the
solution is a power series ?(?) = ∑ ??? ∞ ? ?=0 . Be sure to
clearly indicate the recursion relationship. Simplify as much as
possible. ? ′′ (?) + 9??(?) = 0

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