Ch7: Linear Systems of Differential Equations

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Presentation transcript:

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: System of DE Solutions 2ed order Independent variable: t dependent variables: x, y Example: Order of the system First-order 3ed-order First-order

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: Linear system

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: Matrix Form

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: Homog and Non-homg

IVP Ch7: Linear Systems of Differential Equations High-order System Sec(7.1+7.2): First-order Systems Example: Practical Importance: IVP High-order System Converted Solution First-order System Example: First-order System

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: Example: Transform into first-order system Transform into first-order system

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Example: Consider the first-order linear system of DE (*) Verify that the vector functions are both solutions of (*)

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Def: Therorem ( Existence of a Unique Solution) System of linear first-order DE Matrix Form: There exists a unique solution of IVP(*)

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems Therorem ( Principle of Superposition) Example: (*) Consider the sys of DE: (*) are both solutions of (*) solution of (*) DEF ( Wronskian) Consider the sys of DE: (*) Example: (*) their wronskian is the nxn determinant Find W(X1,X2)

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems THM ( Wronskian) Consider the sys of DE: (*) Example: (*) Linearly dependent or independent ??

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems THM ( general solution for Homog) Consider the sys of DE: (*) The general sol for (*) is Example: Example: (*) Solve IVP (*) Find the general solution for (*)

Ch7: Linear Systems of Differential Equations Sec(7.1+7.2): First-order Systems THM ( general solution for non-Homog) Consider the sys of DE: (*) (**) The general sol for (*) is Example: (*) Example: Solve IVP Sol for Homog Particular sol for non-Homog Find the general solution for (*)

System of Linear First-Order DE How to solve the system of DE System of Linear First-Order DE (constant Coeff) Distinct real Eigenvalues (7.3) repeated real Eigenvalues (7.5) complex Eigenvalues (7.3) Eigenvalue Method System of Linear First-Order DE (Non-homog) Variation of Parameters