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.

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First-order linear equations

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First order linear equations

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

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 method1

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

Example

  • Ex. Solve the equation

  • Sol. An integrating factor is

    Multiplying I(x) to the equation, we get

  • Ex. Solve

  • Sol.


Example1

Example

  • Ex. Solve the equation

  • Sol. Not a linear equation? What if we treat x as dependent

    variable and y as independent variable:


Example2

Example

  • Ex. Solve the equation

  • Sol.

  • Ex. Solve the initial value problem

  • Sol.


Example3

Example

  • Ex. Solve the initial value problem

  • Sol.


Homework 22

Homework 22

  • Section 9.3: 7, 10, 15

  • Section 9.6: 12, 14, 19

  • Page 648: 1


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