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Chapter 4: Higher-Order Differential Equations

Chapter 4: Higher-Order Differential Equations. Chapter 4: Higher-Order Differential Equations. 1. Sec 4.1: Linear DE (Basic Theory). Sec 4.1.1: Initial Value Problem (IVP) Boundary Value Problem (BVP). IVP:. . nth order linear DE. Theroem 4.1 ( Existence of a Unique Solution ).

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Chapter 4: Higher-Order Differential Equations

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  1. Chapter 4: Higher-Order Differential Equations

  2. Chapter 4: Higher-Order Differential Equations

  3. 1 Sec 4.1: Linear DE (Basic Theory) Sec 4.1.1: Initial Value Problem (IVP) Boundary Value Problem (BVP) IVP: . nth order linear DE Theroem 4.1 ( Existence of a Unique Solution) Sol y(x) Exist unique

  4. 1 Sec 4.1: Linear DE (Basic Theory) Theroem 4.1 ( Existence of a Unique Solution) Sol y(x) Exist unique 2 3

  5. Sec 4.1: Linear DE (Basic Theory) Theroem 4.1 ( Existence of a Unique Solution) Sol y(x) Exist unique Find an interval centered about x=0 for which the given IVP has a unique solution 9/p138 2

  6. Sec 4.1: Linear DE (Basic Theory) 2ed order linear DE Problem 1 Problem 2 What is the difference IVP BVP

  7. Sec 4.1: Linear DE (Basic Theory) 2ed order linear DE IVP BVP

  8. Sec 4.1: Linear DE (Basic Theory) 2ed order linear DE IVP BVP Exist and unique When??

  9. BVP can have many, one, or No sol BVP3 BVP2 BVP1 Given that 2-parameter family of solutions unique No sol Infinity number of sol

  10. Sec 4.1.2: Homogeneous Equations diff homogeneous nonhomogeneous 1 (**) is the associated homogeneous DE of (*) 2 Remark: before we solve (*), we have to solve first (**)

  11. Differential Operator Differential Operators

  12. Properties: Differential Operator Linear Operator

  13. Quiz on Monday 2.1 3.1 4.1.1

  14. DE  Differential Operator Form Write as DE where

  15. Homog DE Theroem 4.2 ( Superposition Principle) 1)Constant multiple is sol 2)Sum of two sol is also sol 3) Trivial sol is also a sol ?? are solutions

  16. Homog DE In general Theroem 4.2 ( Superposition Principle)

  17. Linear Dependence & Linear Independence Definition 4.1 IF for every x in I IF not then we say linearly independent Note:Linear Combination Is this set linearly dependent ??

  18. Linear Dependence & Linear Independence Definition 4.1 IF for every x in I IF not then we say linearly independent Is this set linearly dependent ??

  19. Linear Dependence & Linear Independence Definition 4.1 IF for every x in I IF not then we say linearly independent Special case If a set of two functions is lin. Dep, then one function is simply a constant multiple of the other. Is this set linearly dependent ??

  20. Linear Dependence & Linear Independence Definition 4.1 IF for every x in I IF not then we say linearly independent Is this set linearly dependent ??

  21. Linear Dependence & Linear Independence Definition 4.1 IF for every x in I IF not then we say linearly independent Remark A set of functions is linearly dependent if at least one function can be expressed as a linear combination of the remaining Is this set linearly dependent ??

  22. Homogeneous Equations We are interested to find n linearly independent solutions of the homog DE homogeneous

  23. Wronskian Definition 4.2 called the Wronskian of the functions Compute the Wroskian of these functions Compute the Wroskian of these functions

  24. Criterion for Linearly Independent Solutions Theroem 4.3 Linearly Independent These functions are solutions for the DE lin. Indep ?

  25. Fundamental set of solutions Def 4.3 Fundamental set of solutions These functions are solutions for the DE Fund. Set of sol. ? These functions are solutions for the DE Fund. Set of sol. ?

  26. General Solution for Homog. DE Theorem 4.5 Is the general solution for the DE. These functions are solutions for the DE Find the general sol? general sol means what?? Given is a sol for

  27. What is missing How to solve Homog. DE Given a homg DE: Step 1 Find n-lin. Indep solutions Step 2 The general solution for the DE is

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