Log in
HomeClass Notes1,200,000CA670,000

Applied Mathematics 2270A/B Lecture 16: 4.1 Lecture

20 views5 pages
School
Western University
Department
Applied Mathematics
Course
Applied Mathematics 2270A/B
Professor
Alan MacIsaac

For unlimited access to Class Notes, a Class+ subscription is required.

Document Summary

What we"re going to do is we"re going to take a differential equation and transform it. Note: this will yield a new function in terms of s from the original function that is in terms of t. So, this is the definition of the laplace transform. We"re going to find out how to use it in practice. "it"s kind of like when we learned derivatives last year. We always knew that the definition of the derivative was the first principle with the limit as h approaches 0. We developed methods like the product rule and the chain rule in order to take derivatives in practice. That"s what we"re going to be doing with the laplace transform" They"re a short hand way of writing limits. This limit will be evaluated 2 different ways depending on the value of s. It"s not a frequency if t = time. It"s not a temperature or anything like that.

Get access

Grade+
$40 USD/m
Billed monthly
Grade+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
10 Verified Answers
Class+
$30 USD/m
Billed monthly
Class+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
7 Verified Answers

Related Documents

Applied Mathematics 2270A/B Lecture Notes - Lecture 17: Algebraic Equation
harlequinbadger850
Applied Mathematics 2270A/B Chapter Notes - Chapter 1.1: Ordinary Differential Equation
scarlethound445
Applied Mathematics 2270A/B Lecture Notes - Lecture 27: Piecewise
harlequinbadger850

Related Questions

Using a technique of integration and the separation-of-variables technique, you'll discover a function whose antiderivative models the shape of an ideal cable suspended from its endpoints. The function depends on a parameter b, which is determined by the properties of the cable, such as the density of the material out of which it is made.

Using principles from physics, it can be shown that when a perfectly flexible, uniformly dense, and inextensible cable is suspended from its endpoints, then it will assume a shape of a curve y = f(x) satisfying the following second-order differential equation:

where ρ is the density of the material of the cable, g is the acceleration due to gravity, and T is the magnitude of the tension in the cable at its lowest point.

We assume our cable is as in the diagram above and y = f(x) is its shape function; thus, not only does y satisfy the "hanging cable differential equation" above, we also have y '(0) = 0. Hence, making the substitution u =dy/dx in our the hanging cable equation, we obtain the first order, initial-value problem satisfied by u:

where b = ρg/T Solve the preceding initial-value problem, entering your (explicit) solution in the form u = g(x) where g is a function of x in which the constant b is present.

In Section, we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. (See Example 6 in that section) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable.

A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If y(t) is the amount of salt (in kilograms) after t minutes, show that y satisfies the differential equation 

 

Solve this equation and find the concentration after 20 minutes.

 

Single-compartment model: Solve (a), and (b).