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 2.y

yx

( ) ( )y P x y Q x

2xy y x

( ) 0,Q x

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

( ) ( ) (1)y P x y Q x

( )( ( ) ) ( ) ( ) (2)I x y P x y I x Q x

( )( ( ) ) ( ( ) ) (3)I x y P x y I x y ( ( ) ) ( ) ( ),I x y I x Q x

1( ) ( ) ( ) ( ) ( ) .

( )I x y I x Q x dx C y I x Q x dx C

I x

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).

( )( ( ) ) ( ( ) ) ( ) ( ) .I x y P x y I x y I x y I x y ( ) ( ) ( ) (4)I x P x I x

( )( ) .

P x dxI x Ce

( )( )

P x dxI x e

Example Ex. Solve the equation Sol. An integrating factor is

Multiplying I(x) to the equation, we get

Ex. Solve

Sol.

2 23 6 .y x y x 2 33

( ) .x dx xI x e e

3 3 3 32 2 2( 3 ) 6 ( ) 6x x x xe y x y x e e y x e 3 3 3 326 2 2 .x x x xe y x e dx e C y Ce

sin.

y xy

x x

1

( )dxxI x e x

coscos .

C xxy x C y

x

sin ( ) sinxy y x xy x

Example Ex. Solve the equation

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

variable and y as independent variable:

2.

2

dy y

dx x y

22 2.

dx x y xy

dy y y

22( )

dyyI y e y

2 3 12

dxy xy ydy

2 1 2( ) ln | |dy x y y x y C

dy

2( ln | |) .x C y y

Example Ex. Solve the equation

Sol.

Ex. Solve the initial value problem

Sol.

2cos tan 0.dyt y tdt

2sec tan( )tdt tI t e e

tan tan 1.ty Ce t

, (2) 1.2 ln

yy y

y y y x

1

( )dyyI y e y

2

ln .x y yy

Example Ex. Solve the initial value problem

Sol.

2 | |, (1) 1.y y x y

21

1 10 ;

2 4xx y C e x

22

1 10

2 4xx y C e x

21

3(1) 1

4y C e

22 1 2

1 1 3 1(0 ) (0 )

4 4 4 2y y C C C e

2 2

2 2

3 1 1( 0)

4 2 4 .3 1 1 1

( ) ( 0)4 2 2 4

x

x

e x xy

e e x x

Homework 22 Section 9.3: 7, 10, 15

Section 9.6: 12, 14, 19

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