N-th order linear DE

Constant Coeff variable Coeff

Homog NON-HOMOG(find yp)

undeterminatecoefficients

Variation of Parameters

Euler equation In GeneralPower Series

(not in Math-260)

Euler EquationsSec(5.1): page 295 ex: 51-56

Sec(5.3): page 320 ex: 51-58

Sec(5.5): page 347 ex: 57-62

Euler equation

Euler Equation (homog)

( ) 1 ( 1) (1)( 1) 1 0 ( )n n n n

n na x y a x y a xy a y f x

:Example25 '' 3 ' 6 sinx y xy y x

3 29 ''' 4 '' 2 cosx y x y y x

Sec(5.1): page 295 ex: 51-56

Sec(5.3): page 320 ex: 51-58

Sec(5.5): page 347 ex: 57-62

Method-1

Step 2

:Example

or lntx e t x

Method of Solution (homog)

Step 1

Solve: 2 '' 2 ' 4 0x y xy y

1dy dy dt dy

dx dt dx x dt

2

2 2

1 1d y d dy dy

dx x dx dt dt x

2 2

2 2 2

1d y d y dy

dx x dt dt

DE in x,y

Variable coeff

DE in t,y

constant coeff

tx e

1x ty y

x

2

1xx tt ty y y

x

Sec(5.1): page 295 ex: 51-56

Sec(5.3): page 320 ex: 51-58

Sec(5.5): page 347 ex: 57-62

Method-2

Step 2

Step 3

Step 4

:Example

Solve: 24 '' 8 ' 0x y xy y

: y=xmTry

subsitute , ', ''into the DE y y y

Method of Solution (homog)

Step 1Solve: 2 '' 2 ' 4 0x y xy y

2

: , ' ,

'' ( 1) ,

m m

m

Find y x y mx

y m m x

1 2,m mx x1 1, lnm mx x x

cos( ln ), sin( ln )x x x x

Find the values of m

Distinct real

repeated real

No

n-

real

Sec(5.1): page 295 ex: 51-56

Sec(5.3): page 320 ex: 51-58

Sec(5.5): page 347 ex: 57-62

Euler Equation (non-homog)

:ExampleMethod

Step 2

Step 3

Step 1 Solve the homog: cyFind :

pyFind :Use variation of parameters:

The general solution is

pc yyy

32 6'4'' xyxyyx

52 72''' xyxyyx

Sec(5.1): page 295 ex: 51-56

Sec(5.3): page 320 ex: 51-58

Sec(5.5): page 347 ex: 57-62