1. (a) Assume u(0-)-1 V. Find the time at which r(t) = 3 V. (b) Assuming...
Find the general solution of jutt + 2 ut + 2 u 3 u(0,t)ut)-0for all t s o ater for all x E (0, π), t > 0 Be sure to clearly indicate the following steps in your solution: 1. 2. 3. How to use separation of variables How to solve the resulting elgenfuiction/eigenvalue problem How the superposition principle is used.
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject to the nonhomogenoous boundary conditions u(0, t) 1, u(r, t) 0, t>o and the initial condition u(, 0)in() Find the solution u (z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution t) (b) Den ote u(x, t)-u(x, t)-ue(x). Derive the IBVP for the function u(x,t). (c) Find v(x, t) (d) Find u(x,t)
Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
3. In the circuit of Fig. 3, find (t) & it) for > 0. Assume that v(0) = 0 V and i(0)=1 A. Use the time domain method that you learnt in the course to solve the problem. (5 pts) 4u(t) A 222 2. F 1H 64
Find vc(t) for all t> 0, given the following circuit. 10Ω 100 mF vc(t 25(1- u(t) V
3) The switch moves from position a to b at time t-0. Find i(t) for t >0 a 62 i(t) b 3 2 108 V (t 2 H
1. Consider the heat flow problem on the real line, where u(x,t), t > 0 is the temperature at point x at time t: ди 1 a2u t>O (*) at 2 ar2 u(x,0) = sin(7x) = > (a) What is the thermal diffusitivity constant ß? (b) Find the intervals of x where the temparature will increase at t = 0. (c) Sketch the graph of the temperature at t = 0. (d) On the same axes as in (c), sketch...
(b) Consider continuous-time signals xi(t) and x2(t) respectively given as (t +1 -1 <t<o x1(t) = { 2 Ost<2 , I 0 otherwise x2(t) = u(t) – uſt – 2). Find the convolution xı(t) * x2(t). (15 marks)
for_final.pdf Example 9.8-1: Find the complete response v(t) for t>0, assuming the circuit is at steady state at t=0 t=0 4 92 NNN 1H Vs = 6 e-3'u(t) v www (.* v(t)=44/3e2t+1/3e-5t_9e-3t (V) M. Cheng