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  1. J

    X is an accumulation point show there is subsequence that converges to x

    Can we say the sequence is {x-1/k2} for integers>0 ? I'm still a little confused. I know that the interval (x-1/k, x+1/k) will be contained in a neighborhood of x, and thus has infinitely many points. I'm not sure why we are using 1/k though. I guess we are in essence using the definition of...
  2. J

    X is an accumulation point show there is subsequence that converges to x

    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...
  3. J

    Prove x belongs to the set or is an accumulation point.

    So, I might have got it... I tried it by contradiction. Assume x is not a member of S and assume it is not an accumulation point of S. If there is a neighborhood (x - epsilon, x + epsilon) containing x that does not have a point of S, then (x - epsilon) is an upper bound of S that's less than...
  4. J

    Prove x belongs to the set or is an accumulation point.

    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...
  5. J

    Prove 2^(-n) converges

    I was confused how to find it that way. I tried this: 1/2N < e 1/e < 2N ln(1/e) < Nln2 ln(1/e)/ln2 < N does that work?
  6. J

    Prove 2^(-n) converges

    Chose e>0. Let N be any positive integer greater than 1/e. Then, for n>=N we have |1/(2n)-0| < |1/n|= (1/n) <= (1/N) < (1/(1/e)) = e Thus, the sequence converges to 0?? Does that look right?
  7. J

    Prove 2^(-n) converges

    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...
  8. J

    Between any two distinct real numbers there is a rational number

    So, we chose q>1/(y-x)? So then, 0 < p-(x/(y-x)) < 1 0 < p < (y/(y-x)) I'm still lost on how we choose p then.
  9. J

    Between any two distinct real numbers there is a rational number

    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...
  10. J

    Piecewise continuous - step function

    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...
  11. J

    Laplace transform initial value problem

    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...
  12. J

    Laplace transformation t^(3/2)

    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...
  13. J

    Multiple eigenvalue solutions

    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...
  14. J

    Eigenvalue Method

    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
  15. J

    IVP Forced Mechanical vibration

    Yeah, sorry about that!! I was pretty tired when I wrote it I think. x''+2x'+5x=4cos(7t), x(0)=x'(0)=0 Haha. I feel stupid now. Sorry again!
  16. J

    IVP Forced Mechanical vibration

    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)...
  17. J

    Undamped forced oscillations

    x(t)=-2tcos(2t) is not the correct answer. I checked the problem again and the initial conditions, and they are right. I remember in class that to find the A and B of the complimentary solution, you wait to apply the initial conditions until after you find the particular solution as well...
  18. J

    Undamped forced oscillations

    I think you made an error when finding A and B: 4Acos(2t)-4Bsin(2t) = 8sin(2t) ==> A=0 and B=-2? So, x_p = -2tcos(2t)?
  19. J

    Undamped forced oscillations

    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...
  20. J

    Variation of Parameters

    Homework Statement Find a particular solution to: y''+4y=20sec(2t) Homework Equations The Attempt at a Solution y''+4y=0 r^2+4=0 r=+or- 2i So, yc(t) = Asin(2t) + Bcos(2t) yp(t)= -cos(2t) ∫ 10sin(2t)sec(2t)dt + sin(2t) ∫ 10cos(2t)sec(2t)dt = -10cos(2t) ∫...
  21. J

    Free Mechanical Vibrations

    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...
  22. J

    Differential Equation- Undetermined Coefficient

    I already tried that answer as well... It's a no go. There must be a glitch in the system or something...
  23. J

    Differential Equation- Undetermined Coefficient

    I tried that as well: yp=A+Bt+Ct^2+De^(4t) yp'=B+2Ct+4De^(4t) yp''=2C+16De^(4t) 2yp''-yp'-yp= 4C+32De^(4t)-B-2Ct-4De^(4t)-A-Bt-Ct^2-De^(4t) = 27De^(4t)-Ct^2-2Ct-Bt-A-B-4C So, 27De^(4t)=3e^(4t) D= (1/9) -Ct^2=-t^2 C=1 -2Ct-Bt=2t -2(1)-B=2 B=-4 -A-B-4C=0...
  24. J

    Differential Equation- Undetermined Coefficient

    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...
  25. J

    Find the Wronskian

    You're right! The calculation was simple. Thanks!
  26. J

    Second-order IVP

    Hmmm... I integrated it instead of derivating...haha. Well I feel stupid now... Thanks for the help!!
  27. J

    Second-order IVP

    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)=...
  28. J

    Find the Wronskian

    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...
  29. J

    Proove there is an x s.t. x^3+x=6

    Would I need to first prove that it is continuous then?
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