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First-order linear equationsPowerPoint Presentation

First-order linear equations

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First-order linear equations

- A first-order linear equation has the general form
If the equation is called homogeneous; otherwise

it is called inhomogeneous.

- For example, is a linear equation, and an
inhomogeneous one, since it can be written as

Integrating factor method

- To solve the first-order linear equation
we multiply the equation by a suitable function I(x):

If the factor I(x) is chosen such that

then equation (2) becomes

which can be solved by

Integrating factor method

- Thus the key point to solve equation (1) is to find I(x) such
that equation (3) holds true:

This is equivalent to

which is a separable equation for I(x). Its solution is

- Simply taking C=1, we call an integrating
factor of equation (1).

Example

- Ex. Solve the equation
- Sol. An integrating factor is
Multiplying I(x) to the equation, we get

- Ex. Solve
- Sol.

Example

- Ex. Solve the equation
- Sol. Not a linear equation? What if we treat x as dependent
variable and y as independent variable:

Example

- Ex. Solve the equation
- Sol.
- Ex. Solve the initial value problem
- Sol.

Example

- Ex. Solve the initial value problem
- Sol.

Homework 22

- Section 9.3: 7, 10, 15
- Section 9.6: 12, 14, 19
- Page 648: 1

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