Are you looking for the Take Home Exam 1 due Thur Mar 13 ? or the Take Home Exam 2 due Tue May 6 ?

EP Math quals warm-up

1. calculate the inverse of the matrix K=[ A B; 0 D] with A m-by-m and D n-by-n invertible matrices.
2. Find the range (a.k.a. column space) and the null space of the matrix A=[5 7 2; 1 3 2; 3 10 7]
3. Calculate the integral of cos(m x)/cosh(x) from -infinity to + infinity.
   [Hint: use the contour that comes back along z=x+i Pi, enclosing a simple pole at z=i Pi/2]
4. Calculate the integral of sin(x+a) sin(x-a)/(x^2-a^2) from -infinity to + infinity.
   [Hint: 2 sin(x+a) sin(x-a)= cos 2a - cos 2x = Re{ e^(2ia) - e^(2ix)}. Use a contour that goes from -infty on the real axis around the pole at x=-a (the only pole) with a half circle on the top then to infty on the real axis and back to -infty using a large circle. The CLOCKWISE half circle around -a is minus half a residue (you can compute it directly) and show that the integral on the big circle vanishes. See example 15.5 in Greenberg's foundations p 280].

• Chapter 22: 22.9. a, b, c. In each case solve directly but also rewrite as a first order system and solve using the matrix exponential formulation; 22.10; 22.12.a ; 22.13; 22.14. a, c; 22.22; 22.24; 22.27 c, f; 22.28; Use the WKBJ method that was discussed in class to deduce the asymptotic behavior of the solutions to y''=x y for large x; 22.42, 22.43; 22.44; 22.46;

• Chapter 23: 23.2.a, b, c, f, g; 23.3; 23.4; 23.6.c; 23.15; 23.16;
• Chapter 24, 25:
  1. Solve x'=t^2-x^2, with x(0)=0 using Euler's method. Can you find the asymptotic behavior as t-> infinity? (using hints from your numerical results and asymptotics).
  2. Derive a family of Runge-Kutta schemes of order 3 (RK3).
  3. Derive a one-sided (forward) 2nd order finite difference formula for d/dt. Apply the resulting scheme to x'=-x with x(0)=1. Discuss and explain your results.
  4. Calculate the stability diagram for RK2 and RK3. Identify the intersections of the |G|=1 neutral stability boundary with the real and imaginary axes in the "a dt" plane.
  5. Solve x''- a(1-x^2) x' + x=0 with x(0)=1, x'(0)=0, using your own numerical code and whatever numerical scheme you prefer, for a=0.01, 0.1, 1, 10, 100, 1000.