Question

Use the power series method to obtain power series solutions about the given point.

a. y′ = y−x, y(0) = 2, x_{0} = 0.

b. (1+x)y′(x) = py(x), x_{0} = 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

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 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

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?

Without looking for the solutions in a power series around Xo
given, determine the radius of convergence for such solutions in
series for the differential equation:
(x^2 + x-12)y"+(x^2-1)y'+3y=0
and the values for Xo= -5, 0, 6

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

Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity do not differ by an integer. Use the method of
Frobenius to obtain to linearly independent series solutions about
x = 0. Form the general solution on (0, ∞)
3xy”+(2 – x)y’ – y = 0

Use an appropriate infinite series method about x = 0 to find
two solutions of the given differential equation. (Enter the first
four nonzero terms for each linearly independent solution, if there
are fewer than four nonzero terms then enter all terms. Some
beginning terms have been provided for you.)
y'' − xy' − 3y = 0
y1 = 1 + 3/2x^2+...
y2= x +....

Use an appropriate infinite series method about
x = 0
to find two solutions of the given differential equation. (Enter
the first four nonzero terms for each linearly independent
solution, if there are fewer than four nonzero terms then enter all
terms. Some beginning terms have been provided for you.)
y'' − xy' − 3y = 0
y1
=
1
+
3
2
x2 +
y2
=
x
+

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

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