Question

Exercise 7.3.8: In the following equations classify the point x = 0 as ordinary, regular singular, or

singular but not regular singular.

a) x^{2}(1+x2)y′′ +xy=0

b) x^{2}y′′ +y′ +y=0

c) xy′′ +x^{3}y′ +y=0

d) xy′′ +xy′ −e^{x}y=0

e) x^{2}y′′ +x^{2}y′ +x^{2}y=0

Answer #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 =

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

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

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

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.

Find at least one solution about the singular point x = 0 using
the power series method. Determine the second solution using the
method of reduction of order.
xy′′ + (1−x)y′ − y = 0

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

A Non-Constant Coefficient ODE: Solve the
non-constant coefficient ordinary differential equation given
below:
?2?2???2−34?=0,
subject to the boundary conditions (not initial conditions)
?(0)=0,?(1)=1.
After solving, answer the following questions:
i) Is x = 0 in the ODE
a) an anomalous singular point,
b) an irregular singular point,
c) a regulous singularious point.
d) a regular singular point?
ii) If the coefficient “x2” were replaced with
“x3/2” which solution series below would you
use?
a) ?=Σ????∞?=0,
b) ?=Σ??+???+?∞?=0, s a constant,...

X, Y, Z are 3 independent random variables. We know that Y, Z is
the 0-1 random variables indicating whether tossing a regular coin
gets a head (1 means getting a head and 0 means not). We also know
the following equations,
E(X2Y +XYZ)=7
E(XY 2 + XZ2) = 3
Please calculate the expectation and variance of variable X.

Solve the following non homogenous Cauchy-Euler equations for x
> 0.
a. x2y′′+3xy′−3y=3x2.
b. x2y′′ −2xy′ +3y = 5x2, y(1) = 3,y′(1) =
0.

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