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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
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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