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Ch7: Linear Systems of Differential Equations

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.

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Ch7: Linear Systems of Differential Equations

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  1. 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

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

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

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

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

  6. 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

  7. 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 (*)

  8. 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(*)

  9. 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)

  10. 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 ??

  11. 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 (*)

  12. 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 (*)

  13. 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

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