Some mass of material is released at the point z = 0. At t > 0...
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Some mass of material is released at the point z = 0. At t > 0 the material density evolves according to the formula P(-1/t). 1. Sketch p and against z (direct axis x to the right) at some moment t> 0 (no accuracy is required, just show general trends). p=t-/2 exp 2. Verify that p satisfies the diffusion equation 8 p dx t = k 3. Find the flux of material as a function of a and t and show that at x = 1 the material is tranferred to the right and at x = 1 the material is tranferred to the left. A pendulum on a rigid rod oscillates according to the equation 80 t + sin 0 = 0, where is the angle of the pendulum from the vertical. 1. Show that 0 = 0 (the pendulum hanging straight down) and = 7 (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. 2. By linearisation show that the solution tion 0 = 0 is stable; at least, show that it is not unstable. is unstable, while the solu- Some mass of material is released at the point z = 0. At t > 0 the material density evolves according to the formula P(-1/t). 1. Sketch p and against z (direct axis x to the right) at some moment t> 0 (no accuracy is required, just show general trends). p=t-/2 exp 2. Verify that p satisfies the diffusion equation 8 p dx t = k 3. Find the flux of material as a function of a and t and show that at x = 1 the material is tranferred to the right and at x = 1 the material is tranferred to the left. A pendulum on a rigid rod oscillates according to the equation 80 t + sin 0 = 0, where is the angle of the pendulum from the vertical. 1. Show that 0 = 0 (the pendulum hanging straight down) and = 7 (the pendulum balanced vertically up) are possible equilibrium solutions of the equation. 2. By linearisation show that the solution tion 0 = 0 is stable; at least, show that it is not unstable. is unstable, while the solu-
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1 To sketch p and against r at some moment 0 we first note that the density p is given by pt124... View the full answer
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