Consider a curve described by yx from x y -1 0 to x y 1

Question

Math 121A: Final exam-

1. (a) Calculate the eigenvalues and eigenvectors of the matrix

(b) Solve the linear system

2. (a) Calculate the radius of convergence R of the power series

_n=1∑^∞(xⁿ/n(-3)ⁿ).

By considering the series for x = ±R and using appropriate series tests, determine the exact range of x for which it converges.

(b) For the function f(t) = -5 + 2t², calculate f(0), f(1), f(2), and f(3). Use the results to sketch f over the range -3 ≤ t ≤ 3.

(c) Determine the precise set of values of t for which the series

_n=1∑^∞[f(t)]ⁿ/n(-3)ⁿ)

will converge.

3. (a) Let a function f(x) have the Fourier transform f^˜(α). Let g(x) = f(-x) and h(x) = x f(x). Show that the Fourier transforms of g and h are given by g^˜(α) = f^˜(-α) and h^˜(α) = if^˜'(α).

(b) Determine the Fourier transform f^˜(α) of the function

(c) By using the above results, or otherwise, determine the Fourier transform s^˜(α) of s(x) = x²e^-|x|. Show that s^˜(α) is real.

4. (a) Calculate the complex Fourier series f(x) = _n=-∞∑^∞c_ne^inx for the function

defined on the interval -π ≤ x < π.

(b) By evaluating f(π/2), show that

π/4 = 1 - 1/3 + 1/5 - 1/7 + . . . .

5. (a) For two functions f and g that have finite integrals, show that the convolution f ∗ g satisfies

_-∞∫^∞(f ∗ g)(x)dx = (_-∞∫^∞f(x)dx) (_-∞∫^∞g(x)dx).

(b) Consider the functions

Sketch f. Calculate f ∗ g and sketch it. Verify that the result from part (a) holds.

6. (a) Show that y(x) = e^-x and y(x) = x are solutions to the differential equation (x + 1)y'' + xy' - y = 0.

(b) Consider the differential equation

y'' + (xy'/x + 1) - (y/x + 1) = f(x)

for x ≥ 0 subject to the boundary conditions y(0) = 0 and lim_x→∞ y(x) = 0. Calculate a Green function solution of the form

y(x) = ₀∫^∞G(x, x)f(x')dx'.

7. (a) Calculate the Laplace transform F(p) of the function

 (b) Let g(t) satisfy the differential equation

where g(0) = 0 and λ > 0. Calculate an expression for the Laplace transform of the solution, G(p).

(c) Use the Bromwich inversion integral

g(t) = 1/2πi_c-i∞∫^c+i∞e^ptG(p)dp

to calculate g(t), where c is a positive constant.

8. (a) If z = x + iy is a complex number and |z| = r, show explicitly that z¯ = r²/z.

(b) Consider the contour integral

where C(r) is a circle of radius r centered at 0. By using residue calculus or otherwise, evaluate I(r) for the three cases of (i) r > 2, (ii) 1 < r < 2, and (iii) 0 < r < 1.

9. The displacement x(t) of a mass on the end of a damped spring undergoes vibrations of the form

x¨ + 2µx? + kx = 0

where k > 0, and the damping coefficient µ satisfies 0 < µ < √k.

(a) Solve the equation for initial conditions x?(0) = 0, and x(0) = a. (Note: it may help to define q = √(k - µ²) and express your answer in terms of q.)

(b) Let t₀ = 0, and define t_j to be the sequence of successive times when the mass is stationary, so that x?(t_j) = 0. Calculate an expression for t_j. Sketch the curve x(t) over the range t0 ≤ t ≤ t₃ and indicate t₁, t₂, t₃ on your sketch.

(c) For each j, define x_j = x(t_j). By considering |x_j+1 - x_j|, calculate the total absolute distance that the mass covers as it comes to rest.

10. Consider a curve described by y(x) from (x, y) = (-1, 0) to (x, y) = (1, 0). Find the shape of the curve that maximizes the area underneath it, given by

_-1∫¹ y dx,

subject to the constraint that the total arc length of the curve is L. Find the explicit form of y(x) for the cases of L = π and L = π/√2.