In Problems 1–6 solve the wave equation (1) subject to the given conditions.
6.
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- 1arrow_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_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
- 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_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_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_forwardQuestion 2. Which three of these vector functions trace out the same curve? (a) r(t) = (−2+ cos t)i + 9j + (3 — sint)k (b) r(t) = (2 + cost)i - 9j + (−3 – sint)k (c) r(t) = (−2+ cos 3t)i + 9j + (3 − sin 3t)k (d) r(t) = (−2 −— cos t)i + 9j + (3 + sint)k (e) r(t) = (2+ cost)i + 9j + (3 + sin t)karrow_forwardQuestion 2 Express the curve by an equation in x and y given x(t) = 11 – sin(t) and y(t) = cos(t). a) O(x + 11)? + y = 1 b) Ox? + y? = 121 c) x? + (y + 11)² = 1 d) O(x – 11)2 + y² = 1 e) Ox + (v – 11) = 1 ... .arrow_forwardii. Find parametric equations for the Line through (7, 5) and (-5, 7) 7. Calculate dy/dx at the point indicated: f(0) = (7tan 0, cos O), 0=a/4arrow_forwardThe wave equation, subject to the given conditions. problem 6arrow_forward6. Simplified equations for an electric motor can be given O"(t) + 20'(t) = u(t) where 0(t) is the motor shaft angle, and u(t) is the voltage applied to the armature windings. a. Write down a state equation for the motor assuming a state vector x(t) = [0(t) O'(t)] and input u(t). b. Transform the state equation to that for a new state variable z(t) so that the new "A-matrix" is diagonal. c. Assuming that (0) = 0'(0) = 0, solve for x(t), t 2 0, when u(t) = e*, t 2 0.arrow_forwardanswer all questions please.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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