Linear higher-order Differential Equations

1. Linear higher-order Differential Equations Chapter 3

2. Chapter 3:Linear higher-order Differential Equations Overview I. Basic Definitions and Theorems II. Reduction of order III. Homogeneous ODE with constant coefficients IV. Homogeneous Cauchy-Euler ODE V. Method of Undetermined Coefficients (UC) VI. Method of Variation of Parameters (VP)

3. I. Basic Definitions and Theorems Learning Objective At the end of this section you should be able to know - the definitions of linear dependence , linear independence, homogeneous DE, Wronskian, fundamental set of solution. - the basic theorems on existence and uniqueness of solutions of DE, superposition of solutions.

4. I. Basic Definitions and Theorems THEOREM: Existence and uniqueness of solution IVP: If there is an interval containing on which : are continuous is non-zero (different to 0 for any x) Then, a solution of the IVP exists on and is unique.

5. I. Basic Definitions and Theorems Example 1: unique solution on any interval containing 0. 1) Solution : check 2) Uniqueness : the coefficients and the function are continuous. The coefficient of is non-zero on any interval containing 0. The conditions of the previous theorem are then satisfied.

6. I. Basic Definitions and Theorems Example 2: is the only solution on any interval containing All the coefficients of are non-zero. Since the third order equation is linear with constant coefficients, all conditions of existence of a unique solution are satisfied. Hence, is the only solution on any interval containing

7. I. Basic Definitions and Theorems BVP A Boundary value problem can have zero, one or infinitely many solutions

8. I. Basic Definitions and Theorems BVP: Examples is a 2-parameters family of solutions Now, depending on the boundary values, we can have 0, 1 or many solutions 1) No solution

9. I. Basic Definitions and Theorems BVP: Examples 2) One solution 3) 1-parameter family of solutions

10. I. Basic Definitions and Theorems Definition: Homogeneous A linear nth-order DE of the form is said to be homogeneous. Example: is a homogeneous linear second-order DE.

11. I. Basic Definitions and Theorems Superposition principle If are solutions of the homogeneous 2nd order DE on an interval , then the linear combination where , are arbitrary constants, is also a solution on the interval

12. I. Basic Definitions and Theorems Superposition principle Example and are both solutions of the homogeneous linear equation on the interval . By the Superposition principle, the linear combination is also a solution of on

13. I. Basic Definitions and Theorems Definition: Linear dependence A set of functionsis said to be linearly dependent on an intervalif there exist constants not all zero, such that for every in Remark: Linear dependence: One of the functions can be expressed as a combination of the others

14. I. Basic Definitions and Theorems Definition: Linear dependence Example : The set is linearly dependent on Write one of the function as a combination of the others

15. I. Basic Definitions and Theorems Definition: Linear independence A set of functionsis said to be linearly independent on an intervalif the only constants satisfying for any in , are

16. I. Basic Definitions and Theorems Definition: Linear independence Example : The set is linearly independent on If for any real number Then, in particular for for Solution of the system

17. I. Basic Definitions and Theorems Definition: Wronskian Suppose each of the functions possesses at least n-1 derivatives The determinant is called the Wronskian of the functions.

18. I. Basic Definitions and Theorems Theorem: criterion for linearly independent solutions Let be n solutions of the homogeneous linear nth order DE on an interval . The set will be linearly independent if and only if for every in .

19. I. Basic Definitions and Theorems Example 1: is a linearly independent Indeed

20. I. Basic Definitions and Theorems Example 2: is a linearly dependent Indeed

21. I. Basic Definitions and Theorems Exercise I: Determine whether the given set of functions is linearly independent on the interval .

22. I. Basic Definitions and Theorems Definition: Fundamental set of solutions A set of n linearly independent solutions of a nth order homogeneous linear equation on an interval is said to be a fundamental set of solutions on the interval.

23. I. Basic Definitions and Theorems Theorem For any nth-order homogeneous linear equation, there is a fundamental set of solutions on an interval . Theorem : general solution Let be a fundamental set of solutions of a 2nd order homogeneous linear DE on an interval . The general solution of the DE on the interval is where are arbitrary constants.

24. I. Basic Definitions and Theorems Example : are solutions of the DE form a fundamental set of solutions on We conclude is the general solution on

25. I. Basic Definitions and Theorems Definition: Non-Homogeneous A linear nth-order DE of the form is said to be non-homogeneous. Example: is a non-homogeneous. The associated homogeneous linear DE is

26. I. Basic Definitions and Theorems Definition Any solution free of arbitrary parameters, that satisfies the DE is said to be a particular solution Theorem : general solution Let be a fundamental set of solutions the associated homogeneous linear DE on an interval . The general solution of the DE on the interval is where are arbitrary constants.

27. I. Basic Definitions and Theorems Example : Given that is the general solution of the DE Find the general solution of the DE

28. I. Basic Definitions and Theorems Example : We have to find a particular solution of the non-homogeneous equation Take Derive and substitute inside the DE, we obtain General solution

29. II. Reduction of order Learning Objective At the end of this section, we should be able to find a second solution, given a non-trivial solution of a linear second order DE.

30. II. Reduction of order What? Given the linear second order DE and a non-trivial solution on an interval . We search for a solution such that is a linearly independent set of solutions on

31. II. Reduction of order Remark If is a linearly independent set of solutions, then cannot be written as a multiple of . is non constant on

32. II. Reduction of order Example : is solution of the DE on Find an other solution such that is linearly independent on From the previous remark, we write

33. II. Reduction of order Example : solution

34. II. Reduction of order Example :

35. II. Reduction of order Example : Now as , we can just choose then show that they are linearly independent. Compute the Wronskian:

36. II. Reduction of order Example : are linearly independent. general solution

37. II. Reduction of order General method solution on

38. II. Reduction of order General method

39. II. Reduction of order General method

40. II. Reduction of order General method integrate to get

41. II. Reduction of order General method

42. II. Reduction of order Example 2 Given that is a solution of on the interval use reduction of order to find the general solution.

43. II. Reduction of order Example 2 Let

44. II. Reduction of order Example 2

45. II. Reduction of order Example 2 Let

46. II. Reduction of order Example 2

47. II. Reduction of order Example 2

48. II. Reduction of order Example 2 General solution

49. II. Reduction of order Exercise II: Find the second solution if the first solution is given as indicated .

50. III. Homogeneous with constant coefficients Learning Objective At the end of the module, students able to solve the homogeneous linear ODE with constant coefficients using the auxiliary equation.