2. In this question you will find the non-zero separable solutions elar,t-M(r)N(G) of the Klein Gonlon equation 01...
only f and g please 1. The motion of a vibrating string of length T, with fixed endpoints, immersed in a fluid (such as air) can be modeled by Fu 2&u 21 0rT t>0 at Ot2 (P1) u(0,t)u, t) 0 t20 Qu is a damping term, modelling the effect of at where c,>0. The term proportional to air resistance on the string. (a) Explain why the damping term has a minus sign (2 points) (4 points) (b) Consider the separable...
1. The motion of a vibrating string of length , with fixed endpoints, immersed in a fluid (such as air) can be modeled by -27- 0<r<T, t>0 (PI) u(0, t) = u(, t ) t20 0 is a damping term, modelling the effect of at where c,>0. The term proportional to air resistance on the string. (a) Explain why the damping term has a minus sign. (2 points) (4 points) (b) Consider the separable solutions to (P1), ie., those of...
Math Methods, Phys 612/412 Homework 612/6 Problem 3. Riley and Hobson problem #114. Schrödinger's equation for a non-relativistic particle in a constant potential region can be taken as (a) Find a solution, separable in the four independent variables, that can be written in the form of a plane wave, Aexpli(k-i-cut) ψ(z, y, z, t) Using the relationships associated with de Broglie (ji-M) and Einstein (E that the separation constants must be such that hw), show (b) Obtain a different separable...
Question 14 (12 marks) Consider the following separable differential equation. dy cos(z)(-1) dr (a) Find any constant solutions of this differential equation and hence write down the solution with initial value y=- when r=7 (b) Use partial fractions to evaluate 1 dy. 1 (c) Use the method for solving separable differential equations to solve this DE in the case where y 0 when r T. You may assume that the solution does not cross the constant solutions you found in...
2. Find non-trivial (non-zero) product solutions for each of the following homogeneous bound ary value problems: 11(0.1) = 0, 14(r,t) = 0, 1 〉 0 t 〉 0 (a) 14-14x = 0, 0〈x〈L, t > 0 a(0.1) 0, 14(T.t)+γ.11(T,t)-O, t > 0 t>O, γǐs constant. 2. Find non-trivial (non-zero) product solutions for each of the following homogeneous bound ary value problems: 11(0.1) = 0, 14(r,t) = 0, 1 〉 0 t 〉 0 (a) 14-14x = 0, 0〈x〈L, t >...
the DE 3xy', + (2-x)y'-y-0 and when r-0 С.-Cu-1. When r-1. ck-Ser. For Find two power series solutions to the DE. (points 10) 3 k 1,2,3, both k the DE 3xy', + (2-x)y'-y-0 and when r-0 С.-Cu-1. When r-1. ck-Ser. For Find two power series solutions to the DE. (points 10) 3 k 1,2,3, both k
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
PART C ONLY! Thank you. 14. Fix a non-zero vector n R". Lot L : Rn → Rn be the linear mapping defined by L()-2 proj(T), fa TER or all (a) Show that if R", Such that oandj-n -0, then is an eigenvector of L What is its cigenvaluc? (b) Show that is an cigenvector of L. What is its cigenvalue? (c) What are the algebraic and geometric multiplicities of all cigenvalues of L? 14. Fix a non-zero vector n...
3 For the autonomous equation dN/dt NON - 2)(N - 5)2, (a) What are the equilibrium values of N, i.e., the solutions with N(t) constant? (b) Sketch families of solutions in the t-N plane (c) If a solution of the differential equation in this problem has the initial condition NO) 2.1, what happens to that solution after a long time? 3 For the autonomous equation dN/dt NON - 2)(N - 5)2, (a) What are the equilibrium values of N, i.e.,...
$125.00, and r = 5%. Find the Black-Scholes formula for the option paying $10.00 in 3 months if S(T) S Ki or if S(T) 2 K2, and zero otherwise, in the 7. Let S(0) = $100.00, K = $92.00, K2 Black-Scholes continuous-time model. $125.00, and r = 5%. Find the Black-Scholes formula for the option paying $10.00 in 3 months if S(T) S Ki or if S(T) 2 K2, and zero otherwise, in the 7. Let S(0) = $100.00, K...