Question

Consider the following differential equation 32x ^{2}y
'' + 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.

Answer #1

**Solution:**

** **

Without actually solving the differential equation (x^3 - 2x^2 +
5x)y’’ + 4xy’ + 4y = 0, find a lower bound for the radius of
convergence of the power series solutions about the ordinary point
x = 3

2. Without actually solving the differential equation (cos x)y''
+ y' + 8y = 0, find the minimum radius of convergence of power
series solutions about the ordinary point x = 0.
and then, Find the minimum radius of convergence of power series
solutions about the ordinary point x = 1.

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...

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

Find the differential Equation of
xy'-2y+(2x^3)e^-x=0

slove ODE using power series and find the radius of
convergence
(1-x^2)y''-2xy'+2y=e^(2x)

find the minimum convergence radius of the solutions on power
series of the differential equation (x^2 -2x+10)y''+xy'-4y=0
surrounding the ordinary point x=1

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 =

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?

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

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