First-order linear equations - PowerPoint PPT Presentation

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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.

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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