In Problems 1–6 solve the wave equation (1) subject to the given conditions.
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Chapter 12 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
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- (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_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_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_forward
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