In Problems 1–6 solve the wave equation (1) subject to the given conditions.
6.
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- 5C. Under suitable assumptions derive one dimensional wave equation.arrow_forwardShow that the function a(z, t) = bi sin )cos() + b sin cos( 2) Tct 2nct bị sin COS L + b2 sin L COS L where c, L, b1, b2 are nonzero constants with L > 0 and c > 0, is a solution to the one-dimensional wave equation c2.arrow_forwardQuestion 6 Find the unit tangent vector T (0) to the curve C: r (t) = 2 cos(t)i +3 sin(t)j+ 4tk 0, > 1 OT (0) = OT (0) = OT (0) = 3 4. Question 7arrow_forward
- - 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(0) and yu sin(0) + v cos(0).arrow_forwardProblem 2: a. Using the chain rule dy dy dr dr dt dt compute 4 using the parametric equations x = cost, y = sin t, te (-0, 00) Write as a function (i) of t. (ii) of æ and y. (Express the derivative as a rational function, not a piecewise-defined function.) b. Use the chain rule to express in terms of d (dy dt dr dr and dt c. Use the result in part (a) and the chain rule again to compute 4 as a function (i) of t. (ii) of r and y. (Express the second derivative as a rational function, not a piecewise-defined function.) d. Find the derivatives and 4, as in the previous parts, for the parametric equations x = cos 3t, y = sin 3t, te (-0, 0). e. More generally, let f(t) and g(t) be differentiable functions defined over t € (-0, 0). Suppose the curve C1 has parametric equations r = f(t), y = 9(t), te(-∞,00) and the curve C, has parametric equations x = f(2t), y = g(2t), te (-0, a0). Compute both of the derivatives 4 for C1 and for C2, then describe the relationship between these derivatives…arrow_forward(part 1 of 4) Determine fe when 1. fa = 2. fx = 3. fx = 4. fx = 5. fx = 6. fx = 1. fx 2. fx = = f(x, y) 5.x (x + 2y)² 3y (x + 2y)² (part 2 of 4) Determine fe when f(x, y) = x sin(2y − x) + cos(2y — x). = 4x (x + 2y)² = 3x (x+2y)² 4y (x + 2y)² = cos (2y - x) x sin(2y - x) = x cos(2y - x) - sin(2y - x) 2 sin(2y x) x cos(2y - x) - = 5y (x + 2y)² 3. fx = 4. fx = x sin(2y - x) 5. fx = x cos(2y - x) 6. fx = -x sin(2y - x) 7. fx -x cos(2y - x) 8. fx -2 sin(2y - x) - x cos(2y - x) = 2x - Y x + 2y (part 3 of 4) Determine f when f(x, y) = x cos(x + 2y) + sin(x + 2y). 1. fx 2. fx 3. fx 4. fx = x cos(x + 2y) 5. fx 6. fx 7. fx 8. fx = = 2 sin(x + 2y) + x cos(x + 2y) = 2 sin(x + 2y) - x cos(x + 2y) 2x cos(x + 2y) = 2 cos(x + 2y) + x sin(x + 2y) 2 cos(x + 2y) - x sin(x + 2y) -2x sin(x + 2y) -x sin(x + 2y)arrow_forward
- The wave equation 1=10 may be studied by separation of variables: u(x, t) = X(x)T(t). If(x) = -k²X(x), what is the ODE obeyed by T(t)? [] Which of the following solutions obey the boundary conditions X(0) = 0 and X (L) = 0? [tick all that are correct □sin() for & integer sin() sin( (2k+1)mz 2L ) for k integer □ sin(2) sin() □ sin() Which of the following is a possible solution of the above wave equation? ○ cos(kx)e-ket O cos(kex) sin(kt) ○ Az + B ○ cos(kx) sin(kt) O None of the choices applyarrow_forwardProblem 1: (2 marks) Let r(t) = (4sin 21)i + (4cos 2r)j- (e“)k, be the position of particle in space at time . Find the velocity, speed and acceleration of the particle at t=0. Problem 2: (2 marks) A projectile is fired from the origin over the horizontal ground at an initial speed of 900 m/sec and a launch angle of 600 (i) When and how far away will the projectile strike? (ii) How high overhead will the projectile be when it is 9 km downrange. Problem 3: (2 marks) Find the curvature for the curve r(t) (6sin 2t)i + (6cos 2t)j+ 5tk Problem 4: (2 marks) By considering different paths of approach, show that the following function has no limit. 9x*y lim(.y)-(0.0) 7 9x. V5 Problem 5: (2 marks) aff Let f(x, y) = sin(2.x) – y² In(xy- 3). Prove that dxây дудхarrow_forward5. Solve the following Wave Equation: a2U a2u 4 ax2 U(0, t) = U(r, t) = 0 and at2 au U(x,0) = 2sin x + sin 2x, (x, 0) = 0 0arrow_forward25. Find pairs of solutions of Laplace's equation, = 0, based on these complex functions: (a) ze2: (b) z*: (c) z sin z: (d) e cosz. ...****** ********..arrow_forward4. Consider a vibrating string of length L = n that satisfies the wave equation 4- 0 0. Assume that the ends of the string are fixed, and that the string is set in motion with no initial velocity from the initial position u(x,0) = 12 sin 2x – 16 sin 5x + 24 sin 6x. Find the displacement u(r, t) of the string.arrow_forward3. Find the orthogonal trajectory of y =c cos xarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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