2. Monday Class: (next class) Building # 5 PC-Lab 202 How to Solve DE using MATLAB

3. Ch8: System of Linear First-Order Differential Equations Def: Matrix Form: System of linear first-order DE Sec(8.1): System of Linear First-Order Differential Equations

4. Ch8: System of Linear First-Order Differential Equations Solution vector: Verify that : are solutions Initial Value Problem REMARK: Much of the theory of systems of n linear first-order DE is similar to that of linear nth-order DE. Solve: Subject to :

5. Ch8: System of Linear First-Order Differential Equations Initial Value Problem Solve: Subject to : Solve: Subject to : Therorem 8.1 ( Existence of a Unique Solution) There exists a unique solution of IVP(*)

6. Ch8: System of Linear First-Order Differential Equations Therorem 8.2 ( Superposition Principle) Is a solution for (*) are solutions for the homog system (*) Then : are solutions is also solution DEF 8.2 ( Linear Dependence/Independence) Linearly dependent There exist constants not all zeros Otherwise, they are Linearly independent

7. Ch8: System of Linear First-Order Differential Equations DEF 8.2 ( Linear Dependence/Independence) Linearly dependent There exist constants not all zeros Otherwise, they are Linearly independent DEF ( Wronskin)

8. Ch8: System of Linear First-Order Differential Equations DEF ( Wronskin) Find the Wronskin Theorem 8.3 ( Wronskin) Linearly Independent

9. Ch8: System of Linear First-Order Differential Equations Solutions Lin. indep DEF: ( Fundamental Set of Solutions) Solutions Lin. Indepfundamental set of Solutions Theorem 8.5: ( General Solution for HOMOG ) Is the general solution for (*) fundamental set of Solutions Solutions Lin. indep

10. Ch8: System of Linear First-Order Differential Equations Theorem 8.5: ( General Solution for NON-HOMOG ) Is the general solution for (*) Fund. Set for Particular sol Find the general sol for the non-homog system

11. How to solve the system of DE System of Linear First-Order DE (constant Coeff) Distinct real Eigenvalues (8.2.1) repeated real Eigenvalues (8.2.2) Non-real Eigenvalues (8.2.3) System of Linear First-Order DE (Non-homog) Variation of Parameters (8.3.2)