1. Ch7: Linear Systems of Differential Equations
Sec( ): 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( ): First-order Systems
Example:
Linear system

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

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

5. IVP Ch7: Linear Systems of Differential Equations High-order System
Sec( ): 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( ): First-order Systems
Example:
Example:
Transform into first-order system
Transform into first-order system

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

11. Ch7: Linear Systems of Differential Equations
Sec( ): 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( ): 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. 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