6. (20 pts.) The plane y-0 separates region 1 (y 0), which is a dielectric materia with c, -3.5, from region 2 (y < 0), which is free space. If the electric flux density in region 1 is given by D,-15a, +22ay -20a, [nC/m'], find D..
g(t) = | cos(2) 5 if Ost<T if it st<21 if t 2 21 How can we use unit step function to represent it? ag(t) = tult - T) + cos(2t)u(t - 2TT) +5ult - 27) b.g(t) = tu(t) - tu(t - TT) + COS(2t)u(t - T) - COS(2t)u(t - 2T) +5u(t - 2TT) c. g(t)=t + cos(2t)(u(t – T) - ult - 2TT)) +5 O d.g(t)=t-tult - 1) + cos(2t)u(t – ) - cos(2t)u(t - 2T) +5(1 - ult-27)...
Question 2 Find the area of the parallelogram formed by the vectors: U<-44-2> and v<6,7,-2> Round your answer to 2 decimal places and do not type the unit. Question 3 x = - +1 Find the intersection point of the line ( y = 4t - 3 and the plane 4x + y = Z + 2 = 0. z=t-1 The value of t that corresponds to the intersection point is: ti The intersection point is Al
Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1, t> 0 u(x, 0, 1) = 0, uy(x,1,1) = 0, with boundary conditions u(O, y,t) = 0, u(r, y, t) = 0, and initial condition u(x, y,0) = (y – į v?) sin(2x).
Given v (678) = U (,1;8%)*(x",0)de where U (,t;a") = ( 217 )* <im(8=e")°/2nt and 1 V (X",0) = - wie poz' /ħe-x'2/242 (TA)1/4 e’Poz' /H Using Gaussian integrals, show that 1 1/2 -2ħt) 1+ m2 1 6 1410) Lada cara no 15. (24) (-P0t/m)2 Ų (x, t) = eimr/2ht , "ofcas. 771/2 m
(3) for 0 <2<1 u(0,t) = 4,(2, t) = 0; u(,0) = { " 1 for 1 <<< 2 Solve the heat equation and write down the complete solution. You can skip the nonessential steps, but please show the integration.
(1 point) Solve the nonhomogeneous heat problem 24 = 1,+ sin(2.0), 0<I<T, u(0,t) = 0, u1,t) = 0 u(3,0) = 3 sin(4x) uz,t) = sinx, sint Steady State Solution limuz,t) =
find the solution set in interval notation
-44 so +9 -44 b. <0 u2 + 9 -24 c. 20 2 + 9 -u4 d. >0 u? +9 Part 1 out of 4 -u 50 u2 +9 The solution set is
1. Solve the initial-boundary value problem one = 4 for () <<3, t> 0, u(0,t) = u(3, 1) = 0 for t> 0, u(x,0) = 3x – 2” for 0 < x < 3. (30 pts.)
A. : Suppose that u(x, t) satisfies Ut = Uzr +1, € (0,2) u(x,0) = 0 u(0,t) = u(2,t) = 0 Solve for u(x,t). What is lim u(x,t)? B. Consider the heat equation in the region 0 < x < 1, but supoose that the system is heated with a source. This is represented by: Ut = Uzz + cos(2), 1 € (0,7) u(x,0) = 1+ cos(2x) U (0,t) = U7(TT,t) = 0 Solve for u(x, t).