Ch7: Linear Systems of Differential Equations

1
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) 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(7.17.2) First-order Systems
Example
Linear system
3
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) First-order Systems
Example
Matrix Form
4
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) First-order Systems
Example
Homog and Non-homg
5
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) 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(7.17.2) First-order Systems
Example
Example
Transform into first-order system
Transform into first-order system
7
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) 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(7.17.2) 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(7.17.2) 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(7.17.2) First-order Systems
THM ( Wronskian)
Consider the sys of DE
()
Example
()
Linearly dependent or independent ??
11
Ch7 Linear Systems of Differential Equations
Sec(7.17.2) 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(7.17.2) 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