he technique that we used to solve the time-dependent Schrodinger equation in class is known as separation of variables. Use the same technique to investigate solutions of the wave equation: ∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2
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Q: The technique that we used to solve the time-dependent Schrodinger equation in class is known as…
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The technique that we used to solve the time-dependent Schrodinger equation in class is known as separation of variables. Use the same technique to investigate solutions of the wave equation:
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- Evaluate the dP/dT of the equation by keeping V constant, and a & b are temperature-indepent paramaters.Show that cos(ωt − β), cos ωt, sin ωt are linearly dependent functions of t.Consider the order and linearity of the following equations: i. (y+t)dy/dt + y=1 ii. 3dy/dt + ( t+4)y = t2+ d2y/dt2 iii. d2y/dt2 = cos(2ty) Which of the equations is/are of order two and non-linear?
- Suppose that a particle follows the path r(t) = 2 sin(3t) i+4 cos(3t) j. Give an equation (in the form of a formula involving x and y set equal to 0) whose whose solutions consist of the path of the particle. = 0. (Answer in terms of x and y.) Determine the velocity vector of the particle when t = T : v(7) = (Answer in terms of t.) Determine the acceleration vector of the particle whent = t : a(7) = (Answer in terms of t.)A particle's position vector is given by: F(t) = R(1+ cos(wot + q cos wot))& + R sin(wnt + q cos wot)ŷ (= What is the particle's maximum speed? If it helps, you can assume that R, wo, and q are all positive numbers, and that q is very small.Solve the Goursat problem: = 0 Utt- c²Uxx u|x-ct=0 = x² u/x+ct=0 = x² nt: Use the formula for general solutions of wave equation on the real line. =
- Find the wave function and its energy by solving the Schrodinger equation below for the three-dimensional box.The acceleration vector for the spacecraft Dolphin 163 is given by d(t) = (-2 cos(t), 0,–2 sin(t)). It is also known that the velocity and position at t = 0 are ü(0) = (0, V5, 2) and r(0) = (3, 0,0 ). Assume distances are measured in kilometers (km) and time is measured in seconds (s). (a) Find the position function F(t) for the spacecraft. (b) Find the function for the speed of the spacecraft and the speed when t = 0. (c) Compute the curvature of the trajectory when t = 0. (d) At time t = T seconds the spacecraft launches a probe in a direction opposite of N, the unit normal vector to r. If the probe travels along a straight line in the direction it was launched from the spacecraft for 5 km and then stops, what is its resting coordinate?Solve the following wave equation using finite difference method. 4fxx = ftt - Given: f(0, t) = 0 and f(1, t) = 0 f(x, 0) = ft(x, 0) = 0 sin(x) + sin(2x) (Ref: Hyperbolic Equation)
- The acceleration vector for the spacecraft Dolphin 163 is given by a(t) = (-2 cos(t), 0,-2 sin(t)). It is also known that the velocity and position att = 0 are (0) = (0, v5, 2) and F(0) = (3, 0,0 ). Assume distances are measured in kilometers (km) and time is measured in seconds (s). (a) Find the position function F(t) for the spacecraft. (b) Find the function for the speed of the spacecraft and the speed when t = 0. (c) Compute the curvature of the trajectory when t 0. (d) At time t = A seconds the spacecraft launches a probe in a direction opposite of N, the unit normal vector to 7. If the probe travels along a straight line in the direction it was launched from the spacecraft for 5 km and then stops, what is its resting coordinate?Suppose we do not know the path of a hang glider, but only its acceleration vector a(t) = -(3 cos t)i - (3 sin t)j + 2k. We also know that initially (at time t = 0) the glider departed from the point (4, 0, 0) with velocity v(0) = 3j. Find the glider’s position as a function of t.use separation of variables to solve the Laplace equation