Question

7. 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 two linearly independent series solutions about x = 0. Form the general solution on (0, ∞)

2xy”- y’ + y = 0

Answer #1

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

The point x = 0 is a regular singular point of the differential
equation. x^2y'' + (9 /5 x + x^2) y' − 1/ 5 y = 0. Use the general
form of the indicial equation (14) in Section 6.3 r(r − 1) + a0 r +
b0 = 0 (14) to find the indicial roots of the singularity. (List
the indicial roots below as a comma-separated list.) r =

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

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...

Consider the differential equation x^2 y' '+ x^2 y' + (x-2)y =
0
a) Show that x = 0 is a regular singular point for the
equation.
b) For a series solution of the form y = ∑∞ n=0 an
x^(n+r) a0 ̸= 0 of the differential equation about
x = 0, find a recurrence relation that defines the coefficients
an’s corresponding to the larger root of the indicial equation. Do
not solve the recurrence relation.

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

Differential Equation:
Determine two linearly independent power series solutions
centered at x=0.
y” - x^2 y’ - 2xy = 0

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

Consider the following differential equation 32x 2y
'' + 3 (1 − e 2x )y = 0 (b) Determine the indicial
equation and find its roots. (c) Without solving the problem,
formally write the two linearly independent solutions near x = 0.
(d) What can you say about the radius of convergence of the power
series in (c)? (e) Find the first three non-zero terms of the two
linearly independent solutions.

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'' − 2xy' − y = 0

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