CHAPTER 5

1. create or choose a new variable

2. FIRST-ORDER SYSTEMS WITH CONSTANT COEFFICIENTS

2.1. the higher order systems of linear equation can be written in term of first-order system

3. SOME MATRIX ALGEBRA

3.1. design a vector by a single symbol

3.1.1. -

3.1.2. dX/dt=X'=AX

3.2. AX=0 has nontrivial solution.

3.2.1. If det A=0, X not equal to 0

3.3. Any 2x2 matrix multiply with matrix I will get the answer the matrix itself .

3.3.1. AI=A , IA=A

3.4. -

4. SOLUTIONS OF FIRST ORDER SYSTEM

4.1. solve | A-ml |=0

4.1.1. to get eigenvalues

4.2. use distinct eigenvalues

4.2.1. to find eigenvectors

4.3. system is linearly independent

4.3.1. if only if W[X(t)] is not equal to 0

5. New Topic

6. REPEATED EIGENVALUES

6.1. Eigenvalues are real and repeated with the multiplicity of 2

6.2. To find λ : det (A-λI) = 0

6.3. Find it's general solution

7. COMPLEX EIGENVALUES

7.1. When the eigenvalues we get complex number

7.1.1. For example: m=1 + i

7.2. Use Euler's formula

7.2.1. exp[(a+bi)t]=exp[at](cos bt + i sin bt)