Question

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 =

Answer #1

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

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.

Consider differential equation: x3
(x2-1)2 (x2+1) y'' + (x-1) x y' +
y = 0 .. Determine whether x=0 is a regular singular
point. Determine whether x=1 is a regular singular point.
Are there any regular singular points that are complex numbers?
Justify conclusions.

Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note
that this is not a constant coefficient differential equation, but
it is linear. The theory of linear differential equations states
that the dimension of the space of all homogeneous solutions equals
the order of the differential equation, so that a fundamental
solution set for this equation should have two linearly fundamental
solutions.
• Assume that y = x^r is a solution. Find the resulting
characteristic equation for r....

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...

(1 point)
In this problem we consider an equation in differential form
Mdx+Ndy=0Mdx+Ndy=0.The equation
(4e−2y−(20x4y5e−x+2e−xsin(x)))dx+(−(20x5y4e−x+8e−2y))dy=0(4e−2y−(20x4y5e−x+2e−xsin(x)))dx+(−(20x5y4e−x+8e−2y))dy=0
in differential form M˜dx+N˜dy=0M~dx+N~dy=0 is not exact.
Indeed, we have
M˜y−N˜x=

Exercise 7.3.8: In the following equations classify the point x
= 0 as ordinary, regular singular, or
singular but not regular singular.
a) x2(1+x2)y′′ +xy=0
b) x2y′′ +y′ +y=0
c) xy′′ +x3y′ +y=0
d) xy′′ +xy′ −exy=0
e) x2y′′ +x2y′ +x2y=0

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

Differential Equations problem
If y1= e^-x is a solution of the differential equation
y'''-y''+2y=0 . What is the general solution of the differential
equation?

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.

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