2- Methods of Solution to FODE

DEPARTMENT DEPARTMENT OF ENGINEERING SCIENCE COLLEGE OF ENGINEERING AND AGRO-INDUSTRIAL TECHNOLOGY UNIVERSITY OF THE PHILIPPINES LOS BAÑOS

ENSC 21

Mathematical Methods in Engineering Lecture Material

FODE Contains only first derivatives and one independent variable  The dependent/independent variable could be any measurable parameter and may be represented by any symbol 

Types: 1) 2) 3) 4) 5) 6)

Variable Separable Homogenous Equation Exact Equations Linear Equations Bernoulli Equations 2nd Order DE Reducible to FODE

1) Variable Separable → are equations where the terms containing the dependent variable y  and its derivative dy  may be collected in one expression, and the terms containing the independent variable and its derivatives in one expression

 () General Form:  = ℎ() By rearranging, we have:  ℎ    =      + 

Determine the general/particular solution of the following:

1+=0  = − ;  4 = 3 2.   1.

 =   4   4. ′= (− ) ; 3.

 0 =0

2)

Homogenous DE General Form:

 M  x,  y dx   N  x, y dy  0

) ) (, (, Test for Homogeneity: (,) = (,) Case 1:

 y  vx

dy  vdx   xdv

Case 2:

 x  vy

dx  vdy   ydv

(Default)

2)

Homogenous DE If homogenous:

[=; =+] or [=; = +]  whichever is convenient or whichever

1. Let

→

results to a less tedious solution. 2. Rearrange to form a separable equation, integrate 3. Revert back to original

 and  parameters

Determine the general solution of the following: 1. 2. 3.

 +  +2=0 2 +  +  =0 (+24)   84 =0

3)

Exact DE General Form:

 :

 M  x, y dx   N  x, y dy  0

  Test for Exactness:  =   = +      = +    

Determine the general solution of the following: 1. 2.

cossin +(1)=0    +   +sin=0

4)

Inexact DE General Form:

 M  x, y dx   N  x, y dy  0

  Inexact DE:  ≠ 

  ():

- a term or a factor which when multiplied to an inexact DE will make the equation exact .

4)

Inexact DE

1      =  I.    1    II.    =   

  =     

  =     

 = +      = +    

Determine the general solution of the following:

( sin) + ln  cos=0 2.  2x +[   +1 cot 1] = 0 1.

5)

Linear (First Order) DE General Form:

 +    =  ( ) 

  =  () ()= 1   () + 

General Solution

Determine the general solution of the following:

x  4=66 2.  2+  +     1 +  =0  :  =    +   +  3. 1      =13  + 2  1.

6) Bernoulli’s Equation (Special FODE) General Form:

 +=() 

where n ≠ 0, 1

  =   −      − =

1  1        +  ()

General Solution

Determine the general solution of the following: 1. 2. 3.

  + =    + =   =(  1)  : − =  +  + 

7) Second Order Ordinary Differential Equations Reducible to First Order (Special FODE) Case 1: Dependent variable (y ) is missing

    ,  ,  →  ,,   Case 2: Independent variable ( x ) is missing

    ,  ,  →  ,,   

Determine the general solution of the following:

    = 3  2.  ′′=2(  )  3. y  + (  ) =′ 1.