In Problems 1–6 solve the wave equation (1) subject to the given conditions.
2.
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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_forward3. Find the orthogonal trajectory of y =c cos xarrow_forward
- 6. 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_forward4. Solve the wave equation Utt = Uxx, 00 u(0, t) = 0, - u(1, t) = 0 - u(x,0) = x(1 − x), ut(x,0) = 0arrow_forward(a) If V = x³ + axy², where a is a constant, show that +y = 3V əx ду Find the value of a if V is to satisfy the equation a²v__a²v = 0 ду? əx² (b) Show that wave equation is satisfied when a² = b²c² 1 a²u a²u Wave equestion: c² at² əx² u = cos at sin bx (c) Determine the first three non-zero terms of the Taylor series expansion for the given function. f (x) = e2× cos(x) about x =0 (d) The partial differential equation a²u a²u 3D 16 — х2 — 2у for 0 < x < 4, 0 < y< 2 (1.1) ax² ду? is subject to the boundary conditions u(x, 0) = 0 and u(x, 2) = 2(16 – x²) for 0arrow_forwardThe 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_forwardWhich of the following is NOT a possible solution for Laplace's equation? (a) y = (AePx + Be-P*)(Ccos py + Dsin py) (b) y = (Acos px + Bsin px)(CEPY + De PY) (c) y = (Ax + B)(Cy + D) (d) y = (A P* + Be-P*)(CePy + Depy) O a O b O carrow_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_forward2) What is the name of the following equation? a?u a?u ax?' ay? = f(x,y) a) Two-Dimensional Heat Equation b) Two-Dimensional Laplace Equation c) Two-Dimensional Wave Equation d) Two-Dimensional Poisson Equationarrow_forwardChapter 13, Section 13.7, Question 031 Find parametric equations for the tangent line to the curve of intersection of the cylinders x +z = 25 and y + z = 10 at the point (4, –1,3). %3D O x(t) = 12 (t – 4), y (t) = -48 (t + 1), and z(t) = -16 (t – 3) O x(t) = 4, y(t) = -1+ 2t , and z(t) = 3 + 6t O x(t) = 4 + 12t , y(t) = -1 – 48t , and z(t) = -3+ 16t O x(t) = 4 + 3t , y(t) = -1 – 12t , and z(t) = 3 – 4t O x(t) = 4 + 8t , y(t) = -1, and z(t) = 3 + 6tarrow_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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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