Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t > 0. In eachof Problems 9 through 12, find an expression for the temperature u(x, t) if the initial temperature distribution inthe rod is the given function. Suppose that a² = 1.9. u (x,0) = 50, 0

We know that the heat conduction equation is α2uxx=ut Given α2=1 So, the equation becomes uxx=ut…

Answered: Consider the conduction of heat in a…

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 100-L tank was initially empty, but is to be filled with salt-and-water solution. The way the solution is being poured into the tank is through a device the controls the rate at which it is being poured. The controller ensures the solution is being poured at time t (minutes) with the expression (1 + cos(2t)) in liters per minute. It follows that the amount of salt for every liter of the solution being poured at timet (minutes) is described by the expression 0.2(1 + cos(2t)) in kilograms. This implies the rate the salt-and-water solution being poured into the tank alternately gets faster and slower. a) Show how the following differential equation describe the salt concentration s in the solution at any time t. Show full solution ds 0.2(1 + cos 2t)? dt
Convert the following equation to subscript notation and identify the Dependent, Independent Variable, the order, degree, linearity and type of the DE. Show your solution if necessary. 2 Txx + (Tyy)? + Tz = 0 XX
In the previous Problem Set question, we started looking at the position functions (t), the position of an object at time t. Two important physics concepts are the velocity and the acceleration. If the current position of the object at time is as (t), then the position at time h later is a (t+h). The average velocity (speed) during that additional time his (s(t+h)-s(t)) If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h→0. i.e. the derivative a' (t). Use this function in the model below for the velocity function (). h The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a(t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a(t) = ✔ (t) =" (t). Problem set question: A particle moves according to the position functions (t) = etsin (2). Enclose arguments of functions in parentheses. For example, sin…
3. If it takes exactly one minute to drain a 10-gallon tank of water, the volume of water remaining in the tank can be approximated by v(t) = 10(1-t)², where t is time in minutes, 0sts 1. Graph the function. How much time has passed when there is only 2.5 gallons of water left in tank?

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