MAE 82 Engineering Mathematics

Transcription

1 Class otes : First Order Differential Equation on Linear AE 8 Engineering athematics

2 Universe on Linear umerical Linearization on Linear Special Cases Analtical o General Solution Linear Analtical General Solution

3 Introduction First Order on Linear Differential Equations First order differential equations Linear - General Solution on Linear - o general solution Analtical solution eist onl for special classes of non linear equations

4 Introduction Schematics First Order Differential Equations Linear Equation Tpe 1 d a gt d Solution: Integration Factors on Linear Equation Tpe 1 Separable Eqs. d d Solution: Integration Factors Tpe d p t g t d Solution: Integration Factors Tpe Eact Eqs. d d Solution: Integration Factors Tpe Autonomous d f t d Solution: Integration Factors Tpe Convert on Eact to Eact Eqs. Solution 1: Integration Factors Solution : Change of Variables

5 Introduction Relationships Among Classes of Equations Eq. that can be made eact Eact d d Separable d d Autonomous d d f t

6 Tpe 1 - Separable Equations General form of 1 st order DE ote that t was substituted with d d f Rewrite the general form of the first order DE as d d Which is possible to do it b setting f 1

7 Tpe 1 - Separable Equations If it happens that d d d d Separable Equation - terms involving each variable ma be placed on opposite sides of the equation. The Differential Form - suppress the distinction between independent and dependent variables. Solution - Integrating the function and.

8 Tpe 1 - Separable Equations Eample 1 d d 1 d d 1 d 1 d 1 C C

9 Tpe - Eact Equation Definition z is a function of two variables z f Differential of a function of two variables Where the variables have continuous first partial derivative in region R of the - plane. The differential is dz f f d d In the special case when f c f d f d

10 Tpe - Eact Equation - Definition or If a function eists such that Then the left hand side of the equation * becomes an eact differentiation and * can be reduced to * d d d

11 Tpe - Eact Equation Definition - Eample d d * * is an eact eq. 1 1 d d d

12 Tpe - Eact Equation Solution In this case the solution to the differentiation equation is c

13 Tpe - Eact Equation Criteria for Eact Differential The necessar and sufficient condition for the equation d d to be eact is or

14 Tpe - Eact Equation Criteria Eample c d d 5 5 Eact Eq.

15 Tpe - Eact Equation General Solution d d 1 For the eact equation 1 we can write the perfect differential d d d ; If the solution is c Integrating t dt h

16 Tpe - Eact Equation General Solution Differentiating partiall with respect to and using d t dt h d 4 Solving for h b Integrating 4 with respect to h Plug 5 in d d h s ds s t s dtds 5 t dt s ds s t s dtds c

17 Tpe - Eact Equation General Solution - Eample Eample d d ; Check if the differential equation is eact 4 ; 4 eact equation

18 Tpe - Eact Equation General Solution - Eample Solution ethod 1 Solve Comparing Solution c Integrate with Integrate with f g c g c f c

19 Tpe - Eact Equation General Solution - Eample Solution ethod - Solution using the formula See also Eample p.98 Eample p.98 4 dd d d c s c ds dt s t ds s dt t

20 Tpe - Convert on Eact to Eact Diff. Eq. Integration Factors Given d d non eact Can sometimes be reduced to an eact differential equation b multipling the given equation b a function Defined as the integration factor of the differential equation d d Turning it into an eact differential equation 1

21 Tpe - Convert on Eact to Eact Diff. Eq. Integration Factors Case1 : is a function of alone d d a 1 a function of alone d e

22 Tpe - Convert on Eact to Eact Diff. Eq. Integration Factors Case : is a function of alone d d b 1 a function of alone d e

23 Tpe - Convert on Eact to Eact Diff. Eq. Integration Factors Special Cases Eample not eact alone function of case1 1 d d 1 d d e

24 Tpe - Convert on Eact to Eact Diff. Eq. Integration Factors Special Cases Eample ew ew e e e ~ ~ ~ c e ds dt s t ds s dt t ~ Review E. 4 P.1

25 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables Substitution / Change of Variables - Transforming the differential equation into another differential equation b means of substitution Transform the first order differential equation d d f b substituting g u where u is regarded as a function of variable d g d g du d g u gu u d d u d d The differential equation becomes du g u gu u f g u d du d

26 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables Which solve for du d F u If we can determine a solution u than the solution of the original diff eq. is g

27 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables Homogeneous Function Definition - If a function possesses the propert f t t t f for some real number α then f is said to be a homogeneous function of degree α

28 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables Homogeneous Function Eample is a homogeneous function of degree since f f t t t t t t f

29 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables on - Homogeneous Function Eample is a non homogeneous function 1 1 f t t t t t t f on homogeneous 1 f

30 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables A first order differential equation d d is said to be homogeneous if both coefficient function and are homogenous function of the same degree t t t t t t ote: the word homogeneous does not mean the same as it ma also refer to the linear equation a1 a g when g

31 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables If and are homogeneous functions of degree we can also write and using substitution u where u u 1 1 v where v v 1 1 v u v u independent variables

32 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables The equation can be rewritten as or where 1 u d 1 u d 1 u d 1 u d u or u B substituting the differential d ud du into the last equation and gathering terms we obtain a separate differential equation in the variable u and 1 u d 1 u ud du 1 u u1 u d 1 u du

33 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables d 1 u du 1 u u1 u

34 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables Eample d d t t t t t t t tt t t t t Homogeneous function of degree Let Then u d ud du

35 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables After substitution the given equation becomes u d u [ ud du] u u d 1 u d 1 u du Long division Integration 1 u d du 1 u d 1 1 du u Re-substitute u ln 1u ln ln c u

36 Tpe - Convert on Eact to Eact Diff. Eq. Substitution / Change of Variables c ln ln ln1 c ln ln ln c ln ln c 1 ln c ln ce