3.Consider the temperature in a uniform rod of length 2 if the initial temperatureisf (x) = xand if the end x = 0 is insulated while the other end is maintained zero degree tem-perature.(a) Find an initial-boundary value problem of PDE to model the temperature functionu (x, t);(b) Solve this PDE.

Given that the rod of length 2 with an initial temperature of fx=x. At one end, that is at x=0 the…

Answered: Consider the temperature in a uniform…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A particle moves on the x axis with acceleration a(t)=6t-10 m/sec2 for t seconds over [0,10] with s(0)=0 and v(0)=6m/sec. Find the position and velocity at t=1 sec.
2t 14. An object is moving along the x axis with velocity v(t) = x'(t) (1+t2)3/2' starting at the initial position x(0) = 4. (a) Find the position x(t) at time t. (b) Find the position of the object at time t = 4.
A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4. The bar's initial temperature f(x) = 90 degrees Celsius. The ends of the bar (x = 0 and x = 3) are then put in an icy bath and kept at a constant 0 degrees C. Let u(x, t) be the temperature in the bar at x at time t, with t measured in seconds. Find u(x, t) and then uz (2, 0.1). Put u7 (2, 0.1) calculated accurately to the nearest thousandth (3 decimal places) in the answer box.
3. A big slug weighing 10 pounds stretches a spring ½ foot. The slug is removed and replaced by another slug, of mass 1.6 slugs. The slug on the spring is then taken to the bottom of the sea (where the water offers resistance that is approximately equal to the velocity of the object) and then released from a position of 1/3 foot above the equilibrium with an downward velocity of 5/4 feet per second. (a) Set up the DE for this situation and show all the steps to find the position equation for the spring. Use X for the position and t for the time. Round all values to 2 decimal places for simplicity. (b) Graph your equation from (a), labeling the axes, and use your graph to find the second time where the slug is heading downwards and passes equillibrium. Write a sentence to summarize.
2. A tank is initially filled with 100 L of salt solution with a concentration of 0.12 kg/L. At time t = 0, a solution containing 0.2 kg/L of salt is poured into the tank at a rate of 3 L/min. Simultaneously, a drain is opened in the bottom of the tank allowing the solution to leave the tank at a rate of 4 L/min. The tank is well-stirred. Let x(t) represent the amount of salt (in kg) in the tank after t minutes. Find the differential equation and initial condition for which x(t) is a solution. DO NOT SOLVE THE DIFFERENTIAL EQUATION. (t)= S t x' (t) = , x(0) =

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