Download presentation
Presentation is loading. Please wait.
1
Differential Equations
Objective: To solve a separable differential equation.
2
Differential Equations
We will now consider another way of looking at integration. Suppose that f(x) is a known function and we are interested in finding a function F(x) such that y = F(x) satisfies the equation
3
Differential Equations
We will now consider another way of looking at integration. Suppose that f(x) is a known function and we are interested in finding a function F(x) such that y = F(x) satisfies the equation The solutions of this equation are the antiderivatives of f(x), and we know that these can be obtained by integrating f(x). For example, the solutions of the equation are
4
Differential Equations
An equation of the form is called a differential equation because it involves a derivative of an unknown function. Differential equations are different from kinds of equations we have encountered so far in that the unknown is a function and not a number.
5
Differential Equations
Sometimes we will not be interested in finding all of the solutions of the equation, but rather we will want only the solution whose integral curve passes through a specified point.
6
Differential Equations
Sometimes we will not be interested in finding all of the solutions of the equation, but rather we will want only the solution whose integral curve passes through a specified point. For simplicity, it is common in the study of differential equations to denote a solution of as y(x) rather than F(x), as earlier. With this notation, the problem of finding a function y(x) whose derivative is f(x) and whose integral curve passes through the point (x0, y0) is expressed as
7
Differential Equations
Equations of the form are known as initial value problems, and is called the initial condition for the problem.
8
Differential Equations
Equations of the form are known as initial value problems, and is called the initial condition for the problem. To solve an equation of this type, first we will separate the variables, integrate, and solve for C.
9
Example 1 Solve the initial-value problem
10
Example 1 Solve the initial-value problem Separate the variables
11
Example 1 Solve the initial-value problem Separate the variables
Integrate
12
Example 1 Solve the initial-value problem Separate the variables
Integrate Solve for C
13
Example 2 Solve the initial-value problem
14
Example 2 Solve the initial-value problem Separate the variables
15
Example 2 Solve the initial-value problem Separate the variables
Integrate
16
Example 2 Solve the initial-value problem Separate the variables
Integrate Solve for C
17
Example 2 Solve the initial-value problem Separate the variables
Integrate Solve for C
18
Example 3 Solve the differential equation
19
Example 3 Solve the differential equation Separate the variables
20
Example 3 Solve the differential equation Separate the variables
Integrate
21
Example 3 Solve the differential equation Separate the variables
Integrate Solve for y
22
Example 3 Solve the differential equation Separate the variables
Integrate Solve for y
23
Homework Page 364 41, 43 Page 593 15, 17, 27, 29 Section 5.2 43, 45
1, 3, 11, 13
Similar presentations
![1 Fundamental Theorem of Calculus Section 4.4. 2 The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.](/19/5780760/big_thumb.jpg "1 Fundamental Theorem of Calculus Section 4.4. 2 The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.>")
