(3) Solve the wave equation on the half-line: Utt = 16uex, 0< x,t < o %3D with initial conditions u(x, 0) = 3, u(x, 0) = 0, and Dirichlet boundary condition u(0, t) = 0.
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- Q5) Solve the wave equation for vibration of organ pipe subject to the boundary condition: a- u(0,t)=0, t>=0 du(1,1) b- ax - 0,1 20 ди(х.0) -U, cons tant с- d- u(x.0) =0.0<=x<=La²u B/ Solve the wave equation: 01² 3x² Under the condition: u=0 when x = 0 and x = 1 ди = 0 when t = 0 and u(x,0) = x², at a²! a = 1 0 < x < 1.Solve the inhomogeneous wave equation on the real lineUtt − c2Uxx = sin x, x ∈ RU(x, 0) = 0, Ut(x, 0) = 0.Explain what theory you are using and show your full computations.
- Find the solution to the wave equation on half-line: Utt=C²Uzr x > 0, t > 0, u(0, t) = 0, t> 0, u(x,0)=1/x, u₁(x,0) = e, x > 0.a2u satisfies the wave equation əx² -n a²u Verify that U(x, t) = e¬Vkt cos\ax %3D k at2Let f(x,t)=cos(12x+4t). Find the value of K so that f satisfies the wave equation ∂^2f/∂x2=K∂^2f/∂t^2
- 8) Find the position vector r(t) for a particle with acceleration a(t) = (5t, 5 sin t, cos 6t), initial velocity (0) = (3, -3, 1) and initial position (0) = (5, 0, -2).= Use variables separation method to solve the wave equation uxxutt. This function is defined on spatial domain 0 0. Subject to boundary conditions: ux(0, t) = u,(a, t) = 0 and initial conditions: u(x, 0) = 0 and u₁(x,0) = f(x)Find the general solution to the following 3-by-3 linear system: -5 10 8 X. dx 2 -4 dt 3 -5 8
- Q.4 Solve the wave equation: U =U +t+1- 1) =5, U(7,t) = cost U(x,0) = , U,(x,0) =for wave equation, seperation of vairables u(x,t)=X=(x)T(t)Q5. a) Find u(x, 1) from the wave equation, where length of string is L=1, c² = 1 and the initial velocity is zero and the initial deflection is as follows. (CLO-2, PLO-3) 1/4 1/4 3/4 -1/4. b) Find the temperature u(x,t) from the heat equation in a laterally insulated copper bar 80 cm long with c2=1.158 cm2/sec. If the initial temperature is represented by the below graph and the ends are kept at 0°C. (CLO-2, PLO-3) f(x) 1 1/2 y=x^2 + x 1/2 1. 3/2