Question

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"+k^{2}x^{2}y=0, x_{0}=0,
k-constant

Answer #1

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

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

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)

solve the following differential equations or initial value
problems answers may be left as implicit solutions (or in terms of
an integral).
2yy'+1=y^2 + x y(0)=1

solve the following differential equations or initial value
problems answers may be left as implicit solutions (or in terms of
an integral).
y'=5y+e^(-2x)y^(-2) y(0)=2

solve the following differential equations or initial value
problems answers may be left as implicit solutions (or in terms of
an integral).
yy'+x=(x^2+y^2)^1/2

Find two solutions of a power series for the differential
equation y'' - xy = 0 surrounding the ordinary point x=0

Series Solutions Near a regular singular point:
Find two linearly independent solutions to the given differential
equation.
3x2y"-2xy'-(2+x2)y=0

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.

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