Homework Statement
Suppose x is an accumulation point of {an: n is a member of integers}. Show there is a subsequence of (an) that converges to x.
The Attempt at a Solution
I'm a little stuck on this one. I know that since x is an accumulation point then every neighborhood around x...
Homework Statement
Let S be a nonempty set of real numbers bounded from above and let x=supS. Prove x either belongs to the set or is an accumulation point of S.
Homework Equations
x is an accumulation point of S iff each neighborhood of x contains a member of S different from x. That...
Homework Statement
Using the definition of convergence to prove that the sequence {2^(-n)} converges
Homework Equations
The Attempt at a Solution
So, I just don't think I am thinking straight or something. Here is what I got so far:
Chose e>0. Let N be any positive integer...
Homework Statement
Let x and y be real numbers with x<y and write an inequality involving a rational
number p/q capturing what we need to prove. Multiply everything in your inequality by q,
then explain why this means you want q to be large enough so that q(y-x)>1 . Explain
how you...
Homework Statement
Consider the following initial value problem:
y''+4y = 9t, 0<=t<2
............0, t>=2
Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)
Homework Equations
The Attempt at a Solution
I am just having trouble with the...
Homework Statement
Use the Laplace transform to solve the following initial value problem:
x' = 7 x + 5 y, y'= -2 x + e5t, x(0)=0, y(0)=0
Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for Y(s) and X(s)
Homework Equations...
Homework Statement
Just a quick question concerning a Laplace transformation...
Find the Laplace transform of the following function:
f(t)=10t3/2-e(-7t)
Homework Equations
The Attempt at a Solution
I wasn't sure what to do with the t3/2 so I just followed the formula for t1/2...
Homework Statement
Solve the system.
dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2]
Homework Equations
The Attempt at a Solution
I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors:
(1-λ)(-7-λ)+16=0
λ2+6λ+9=0...
Homework Statement
solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]
Homework Equations
The Attempt at a Solution
I know I need to find the Eigenvalues but then I get a little confused from there.
(λ-3)(λ+3)=0
λ=3, -3
Homework Statement
The solution to the Initial value problem, x''+2x'+5x=0, is the sum of the steady periodic solution x_sp and x_tr. Find both.
Homework Equations
The Attempt at a Solution
I already found x_sp ( the particular solution). It is (-44/533)cos(7t)+(14/533)sin(7t)...
Homework Statement
This is an example of an Undamped Forced Oscillation where the phenomenon of Pure Resonance Occurs.
Find the solution of the initial value problem:
x'' + 4 x = 8 sin(2 t) , x(0)=x'(0)=0
Homework Equations
The Attempt at a Solution
in class we were given...
Homework Statement
A mass m=4 is attached to both a spring, with spring constant k=37, and a dash-pot with damping constant c=4.
The ball is started in motion with initial position x0=1 and initial velocity v0=8 .
Determine the position function x(t).
Homework Equations
The...
Homework Statement
Find a particular solution to the differential equation
2 y'' - 1 y' - 1 y = -1 t^2 + 2 t + 3 e^{4 t} .
Homework Equations
The Attempt at a Solution
I have attempted this problem many times. I think I am having trouble assuming what the general form is...
Homework Statement
Find y as a function of t if 36y''-132y'+121y=0, y(0)=5, y'(0)=4
The Attempt at a Solution
36y''-132y'+121y=0
36r^2-132r+121=0
(6r-11)^2
So, general solution
y(x) = C1*e^(11x/6)+C2*x*e^(11x/6)
y'(x)=(11/6)*C1*e^(11x/6)+C2*e^(11x/6)*((6x-11)-(36/121))
y(0)= C1=5
y'(0)=...
Homework Statement
Find the Wronskian W(t)=W(y1,y2) where I have found y1=1 and y2=(2/9)-(2/9)e^(-9t/2)
The Attempt at a Solution
I am not sure how to do the Wronskian. We haven't talked about at all in class and I am not even sure what exactly it does. Any help would be greatly...
Homework Statement
Show that there is an x∈R such that x^3+x=6.
The Attempt at a Solution
I'm not exactly sure where to get started with this proof. I think I would need to define a set S={x∈R: x>0 and x^3+x≤6}. Assume S is bounded, and then find lub(S)...?