Chapter 6 Differential Equations.pdf

Eulers Method Eulers Method

Eulers Method Eulers Method

________- a numerical approach to approximating the particular solution of the differential equation 𝑦𝑦 ′= 𝐹𝐹 (𝑥𝑥, 𝑦𝑦) that passes through the point �𝑥𝑥0, 𝑦𝑦0�.

xy plane

At each point (x, y) in the ________ where F is defined, the differential equation determines the slope 𝑦𝑦 ′= 𝐹𝐹 (𝑥𝑥, 𝑦𝑦) of the solution at that point.

Theorem

________: Exponential Growth and Decay Model If y is a differentiable function of t such that y> 0 and y= ky for some constant k, then.

dx

All x terms can be collected with ________ and all y terms with dy, and a solution can be obtained by integration.

Logistic Differential Equation

________ Carrying Capacity- the maximum population y (t) that can be sustained or supported as time t increases.

Slope Fields

________ Consider the differential equation 𝑦𝑦 ′= 𝐹𝐹 (𝑥𝑥, 𝑦𝑦) where 𝐹𝐹 (𝑥𝑥, 𝑦𝑦) is some expression in x and y.

General Solutions

________- the form in which every solution of a differential equation shows upon.

Decay Models

Growth and ________ The rate of change of a variable y is proportional to the value of y.

Theorem

________: Solution of a First- Order Linear Differential Equation An integrating factor for the first- order linear differential equation.

6.1 Slope Fields and Eulers Method General and Particular Solutions Differential equation

is an equation that involves x, y, and derivatives of y

Solution of a differential equation

a function that satisfies y = f(x) when y and is derivatives are replaced by f(x) and its derivatives

General Solutions

the form in which every solution of a differential equation shows upon

Particular solution

are solutions that cannot be written as special cases of general solutions

Eulers Method Eulers Method

a numerical approach to approximating the particular solution of the differential equation 𝑦𝑦 ′ = 𝐹𝐹(𝑥𝑥, 𝑦𝑦) that passes through the point �𝑥𝑥0, 𝑦𝑦0�

Identify the starting point

�𝑥𝑥0, 𝑦𝑦0� & slope 𝐹𝐹�𝑥𝑥0, 𝑦𝑦0�

6.2 Differential Equations

Growth and Decay Differential Equations Solving a Differential Equation

Theorem

Exponential Growth and Decay Model If y is a differentiable function of t such that y > 0 and y = ky for some constant k, then

Logistic Differential Equation Carrying Capacity

the maximum population y(t) that can be sustained or supported as time t increases

It is also called the upper limit L. Logistic Differential Equation

used to describe the growth of a population

Theorem

Solution of a First-Order Linear Differential Equation An integrating factor for the first-order linear differential equation