- §7.2, §7.3: Linear Algebra Review
- Solving a system of equations with augmented matrices
- ...even if the system is underdetermined (many solutions)

- Definition of linear independence
- Checking for linear independence using the Wronskian determinant
- Eigenvalues & eigenvectors: definition and how to find them

- Solving a system of equations with augmented matrices
- §7.1, §7.4: Basic Theory of Systems of Differential Equations
- Any nth order differential equation can be rewritten as a system of n 1st order equations
- Meaning of linear vs. nonlinear & homogeneous vs. nonhomogeneous for systems
- A system of linear differential equations can be written as a single matrix equation
- A homogeneous system of n equations should have exactly n independent solutions
- Any linear combination of these solutions is also a solution

- §7.5: Solving Homogeneous Linear Systems with Constant Coefficients
- Can be written as
**x**' =**Ax** - To solve, find eigenvalues and eigenvectors of
**A**- If r is an eigenvalue with eigenvector
**ξ**, then e^{rt}**ξ**is a solution - General solution:
**x**= c_{1}e^{r1t}**ξ**^{(1)}+ c_{2}e^{r2t}**ξ**^{(2)}+ c_{3}e^{r3t}**ξ**^{(3)}+ ... + c_{n}e^{rnt}**ξ**^{(n)}

- If r is an eigenvalue with eigenvector
- Specific solutions can be found by substituting
**x**(0) for**x**and 0 for t- ...then solve the resulting system for c
_{1}, c_{2}, ..., c_{n}

- ...then solve the resulting system for c

- Can be written as
- §7.6: Complex Eigenvalues
- Solution is still
**x**= c_{1}e^{r1t}**ξ**^{(1)}+ c_{2}e^{r2t}**ξ**^{(2)}+ c_{3}e^{r3t}**ξ**^{(3)}+ ... + c_{n}e^{rnt}**ξ**^{(n)} - If r is complex, use Euler's Formula to expand: e
^{(a+bi)t}= e^{at}·(cos bt + i·sin bt) - Combine entire general solution into one vector (e
^{at}can remain outside) - Split into a vector of real terms plus a vector of imaginary terms
- Factor (c
_{1}+ c_{2}) out of the real vector and (c_{1}- c_{2})i out of the imaginary vector - Replace (c
_{1}+ c_{2}) and (c_{1}- c_{2})i with C_{1}and C_{2}for pure real solution

- Solution is still
- §7.8: Repeated Eigenvalues
- A repeated eigenvalue isn't a problem as long as it produces distinct eigenvectors
- Repeated eigenvectors, however, call for a different approach
- Suppose r is the (only) eigenvalue and
**ξ**is the (only) eigenvector - Clearly one of the solutions is e
^{rt}**ξ** - The other solution is te
^{rt}**ξ**+ e^{rt}**η**, where**η**is another vector s.t.**Aη**= r**η**+**ξ** - To find
**η**, solve (**A**- r**I**)**η**=**ξ**with an augmented matrix (you know r and**ξ**already) **η**can be written as a multiple of**ξ**plus another vector; the multiple of**ξ**can be ignored- General solution:
**x**= c_{1}e^{rt}**ξ**+ c_{2}(te^{rt}**ξ**+ e^{rt}**η**)