CHAPTER 5

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CHAPTER 5 by Mind Map: CHAPTER 5

1. create or choose a new variable

2. REPEATED EIGENVALUES

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

2.2. To find λ : det (A-λI) = 0

2.3. Find it's general solution

3. FIRST-ORDER SYSTEMS WITH CONSTANT COEFFICIENTS

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

4. SOME MATRIX ALGEBRA

4.1. design a vector by a single symbol

4.1.1. -

4.1.2. dX/dt=X'=AX

4.2. AX=0 has nontrivial solution.

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

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

4.3.1. AI=A , IA=A

4.4. -

5. COMPLEX EIGENVALUES

5.1. When the eigenvalues we get complex number

5.1.1. For example: m=1 + i

5.2. Use Euler's formula

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

6. SOLUTIONS OF FIRST ORDER SYSTEM

6.1. solve | A-ml |=0

6.1.1. to get eigenvalues

6.2. use distinct eigenvalues

6.2.1. to find eigenvectors

6.3. system is linearly independent

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

7. New Topic