Question

Fix a ∈ C and b ∈ R. Show that the equation
|z^{2}|+Re(az)+b = 0 has a solution if and only if
|a^{2}| ≥ 4b. When solutions exist show the solution set is
a circle. (proof in the form of a generic If and only if
proof).

Answer #1

(Linear Algebra) Consider the difference equation.
yk+2 - 4yk+1 + 4yk = 0, for all
k
(a) After using auxiliary equation, the solutions have the form
rk and k(rk). Find the root, r, and show that
yk = k(rk) is a solution.
(b) Show that rk and k(rk) are linearly
independent and form the general solution of the difference
equation.

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable.
Then f and g differ by a constant if and only if f ' (x) = g ' (x)
for all x ∈ [a, b].
b) For c > 0, prove that the following equation does not have
two solutions. x3− 3x + c = 0, 0 < x < 1
c) Let f : [a, b] → R be a differentiable function...

Determine the values of r for which the differential equation
y′′′+9y′′+20y′=0 has solutions of the form y=e^rt. Enter the values
of r in increasing order. If there is no answer, enter DNE.
r = ___
r = ___
r = ___

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

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.

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

Show that the equation
x3 − 18x +
c = 0
has at most one root in the interval
[−2, 2].

let A (1 , 2, -3), B (2, 1, 4) and C (0, 0, 2) be three points
in R^3
a) give the parametric equation of the line orthogonal to the
plane containing A, B and C and passing through point A.
b) find the area of the triangle ABC
linear-algebra question
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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...

In Euclidean (flat) space, a circle’s circumference C is related
to its radius r by C = 2πr. Formally, this can be shown by
integrating the infinitesimal distance element ds around the
circumference:
C = ! ds = ! 2π 0 rdθ = 2πr , (1)
where the second equality holds because the 2-D Euclidean metric
is ds2 = dr2 + r2dθ2, and dr = 0 for a circle of constant radius
r.
(a) In a 2-D space of positive...

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