Engineering Mathematics Ⅰ

1. Engineering Mathematics Ⅰ 呂學育 博士 Oct. 13, 2004

2. 1.5 Integrating Factors • The equation is not exact on any rectangle. Because and and  =The equation is not exact on any rectangle.

3. 1.5 Integrating Factor • Recall • Theorem 1.1 Test for Exactness is exact on if and only if (), for each in ,

4. 1.5 Integrating Factor If is exact, then there is a potential function and and (*)  implicitly defines a function y(x) that is general solution of the differential equation. Thus, finding a function that satisfies equation(*) is equivalent to solving the differential equation.

5. 1.5 Integrating Factor • Definition 1.5 Let and be defined on a region of a plane. Then is an integrating factor for if for all in , and is exact on .

6. 1.5 Integrating Factor • Example 1.21 is not exact. Here and For to be an integrating factor, 

7. 1.5 Integrating Factor • Example 1.21 For to be an integrating factor,   To simplify the equation, we try to find as just a function of   This is a separable equation.

8. Recall 1.2 Separable Equations • A differential equation is called separable if it can be written as • Such that we can separate the variables and write • We attempt to integrate this equation

9. 1.5 Integrating Factor • Example 1.21  Integrate to obtain  to get one integrating factor  (*) The equation (*) is exact over the entire plane, FOR ALL (x,y) !

10. Recall 1.5 Integrating Factor If is exact, then there is a potential function and and (*)  implicitly defines a function y(x) that is general solution of the differential equation. Thus, finding a function that satisfies equation(*) is equivalent to solving the differential equation.

11. 1.5 Integrating Factor • Example 1.21 and Then we must have  The general solution of the original equation is 

12. 1.5.1 Separable Equations and Integrating Factor • The separable equation is in general not exact. Write it as and In general However, is an integrating factor for the separable equation. If we multiply the DE by ,we get an exact equation. Because

13. Recall 1.3 Linear Differential Equations • Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. • Non-linear: Differential equations that do not satisfy the definition of linear are non-linear. • Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

14. Recall 1.3 Linear Differential Equations • Example 1.14 is a linear DE. P(x)=1 and q(x)=sin(x), both continuous for all x. An integrating factor is Multiply the DE by to get Or Integrate to get The general solution is

15. 1.5.2 Linear Equations and Integrating Factor • The linear equation Write it as and so the linear equation is not exact unless However,  is exact because

16. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.1 Homogeneous Differential Equations Definition 1.6.1 Homogeneous Equation A first-order differential equation is homogeneous if it has the form For example: is homogeneous while is not.

17. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.1 Homogeneous Differential Equations A homogeneous equation is always transformed into a separable one by the transformation  and write Then becomes   And the variables (x, u) have been separated.

18. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.1 Homogeneous Differential Equations Example 1.25  Let y=ux  or •  • the general solution of the transformed equation • the general solution of the original equation

19. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Bernoulli Equations A Bernoulli equation is a first-order equation (*) in which is a real number. If or separable and linear ODE Let for , then (**) (*)  (*),(**) linear ODE

20. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Bernoulli Equations Example 1.27 which is Bernoulli with , and Make the change of variables , then and so the DE becomes upon multiplying by  a linear equation

21. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Bernoulli Equations Example 1.27 • a linear equation • An integrating factor is •  • Integrate to get  • The general sol of the Bernoulli equation is

22. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Riccati Equations Definition 1.8 A differential equation of the form is called a Riccati equation A Riccati equation is linearly exactly when Consider the first-order DE If we approximate ,while x is kept constant,  How to transform the Riccati equation to a linear one ?

23. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Riccati Equations How to transform the Riccati equation to a linear one ? Somehow we get one solution, , of a Riccati equation, then the change of variables transforms the Riccati equation to a linear one.

24. 1.6 Homogeneous, Bernoulli, and Riccati Equations 1.6.2 The Riccati Equations Example 1.28 By inspection, , is one solution. Define a new variable z by the change of variables Then  Or a linear equation