Apr 15, 2002
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- Introduction to differential equations
	- closed-form solution versus
	  numerical solutions

- Sample problems and solutions
	- x' = x (x = ce^t)
	- x' = -x (x = ce^(-t))
	- x'' = x (x = ce^t + de^(-t))
	- x'' = -x (x = c sin(t) + d cos(t))
	- x' = 2x (x = ce^(2t))

- In general, there exist a family of solutions
	- if initial value is given, then we can
	  pin point a specific function

- Initial value problems
	- e.g., x' = 2x, x(0) = 15
	- answer here is x = 15 e^(2t)

- Solving differential equations by Taylor series
	- first-order approximations (Euler method)

- Taylor series methods of higher orders
	- how to get expression for higher order 
	  derivatives?

- Causes of errors in differential equations 

Apr 17, 2002
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- Causes of errors in differential equations (contd.)
	- roundoff error
	- truncation error
	- error in recording values (perturbations of initial
	  value)

- Better methods
	- Runge-Kutta methods (e.g., order 2)

Apr 19, 2002
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- Stability of differential equations to numerical solution
	- convergence of solution curves
	- divergence of solution curves
	- sufficient tests for stability

- Solving a system of ODEs
	- coupled
	- uncoupled

- Solving higher-order ODEs
	- introduce new variables to reduce to first-order system