Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Systems of Differential Equations In the introduction to this section we briefly discussed how a system of differential equations can arise from a population problem in which we keep track of the population of both the prey and the predator.
We shall have to find a new approach to solving such an equation. Moreover, the fact that we can obtain such a solution analytically will prove very useful when we investigate more complicated equations and systems of equations.
For example, in a hard rock mining operation, ore is often pulverized and then processed using chemicals to extract certain minerals of value.
Soft rock mining operations such as coal mining or extracting oil from tar sands might use solvents or water to extract any commodity of value. The material that is left over after the minerals, coal, or oil is extracted can often present huge environmental challenges.
There are different ways of processing mine tailings, but one way is to store them in a pond, especially if water is used in the mining operation. This method allows any particles that are suspended in the water to settle to the bottom of the pond. The water can then be treated and recycled.
Suppose that we have a gold mining operation and we are storing our tailings in a pond that has an initial volume of 20, cubic meters. When we begin our operation, the tailings pond is filled with clean water. The pond has a stream flowing into it, and water is also pumped out of the pond.
Chemicals are used in processing gold ore.
These chemicals such as sodium cyanide can be highly poisonous and dangerous to the environment, and the water must be treated before it is released into the watershed. Suppose that cubic meters per day flow into the pond from stream and cubic meters are pumped from the pond each day to be processed and recycled.
Thus, the water level of the pond remains constant. We will assume that water in our tailings pond is well mixed so that the concentration of chemicals through out the pond is fairly uniform.
In addition, any particulate matter pumped into the pond from the stream settles to the bottom of the pond at a rate of 50 cubic meters per day. Thus, the volume of our tailings pond is reduced by 50 cubic meters each day, and our tailings pond will become full after days of operation.
We shall assume that the particulate matter and the chemicals are included in the cubic meters that flow into the pond from the stream each day. We wish to find a differential equation that will model the amount of chemicals in the tailings pond at any particular time.
In fact, it is not even separable.
We will have to use a different approach to find a solution. However, the amount of chemicals decreases as the pond begins to fill with sediment.
We can model how petroleum products are mixed together in a refinery, how various ingredients are mixed together in a brewery, or how greenhouse gases move across various layers of the earth's atmosphere.
A brine mixture containing one pound of salt per gallon flows into the top of the tank at a rate of 5 gallons per minute. A well mixed solution leaves the tank at rate of 4 gallons per minute.
We wish to know how much salt is in the tank, when the tank is full. We can use Sage to easily check the solution of our initial value problem.A system of first order linear ordinary differential equation can be expressed as the following form or in the matrix form where the matrix contains only constants and is function of.
Sturm–Liouville theory is a theory of a special type of second order linear ordinary differential equations. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined in terms of .
Nov 10, · Solves a system of two first-order linear odes with constant coefficients using an eigenvalue analysis.
The roots of the characteristic equation are real and distinct. Sturm–Liouville theory is a theory of a special type of second order linear ordinary differential equations. Their solutions are based on eigenvalues and corresponding eigenfunctions of linear operators defined in terms of second-order homogeneous linear equations.
Example #5 – solve the Linear First-Order Differential Equation given an Initial Condition Example #6 – solve the Piecewise Linear First-Order Differential Equation Solving Exact ODEs. 1 – 3 Convert each linear equation into a system of first order equations.
1. y″ − 4y′ + 5y = 0 2. y″′ − 5y″ + 9y = t cos 2 t 3.
y(4) + 3y″′ − πy″ + 2πy′ − 6 y = 11 4. Rewrite the system you found in (a) Exercise 1, and (b) Exercise 2, into a matrix-vector equation. 5.