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SAGE can define piecewise functions like $\begin{array}{ll} x \ {\mapsto}\ \sin \left( \frac{{\pi \cdot x}}{2} \right) &\text{on $(0, 1)$}\\ x \ {\mapsto}\ 1 - {\left( x - 1 \right)}^{2} &\text{on $(1, 3)$}\\ x \ {\mapsto}\ -x &\text{on $(3, 5)$} \end{array}$ as follows: {{{ sage: f(x) = sin(x*pi/2) sage: g(x) = 1-(x-1)^2 sage: h(x) = -x sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: latex(P) }}} |
Differential Equations
First order DEs
IVPs, Direction Fields, Isoclines
Direction Fields, Autonomous DEs
Separable DEs, Exact DEs, Linear 1st order DEs
Numerical method: Euler (or Constant Slope)
Applications (Growth/Cooling/Circuits/Falling body)
Higher order DEs
IVPs/General solutions, Basic theory
Numerical methods for higher order DEs
Constant coefficient case: Undetermined Coefficients
Application: springs (free, damped, forced, pure resonance)
Application: Electrical Circuits
Laplace Transform (LT) methods
Inverse Laplace & Derivatives
1st Translation Thrm
Partial Fractions, completing the square
Unit Step Functions
SAGE can define piecewise functions like $\begin{array}{ll}
x \ {\mapsto}\ \sin \left( \frac{2} \right) &\text{on (0, 1)}\\ x \ {\mapsto}\ 1 - {\left( x - 1 \right)}^{2} &\text{on (1, 3)}\\ x \ {\mapsto}\ -x &\text{on (3, 5)}
\end{array}$
as follows:
sage: f(x) = sin(x*pi/2) sage: g(x) = 1-(x-1)^2 sage: h(x) = -x sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: latex(P)
2nd Translation Theorem
Derivative thrms, Solving DEs
Convolution theorem
Dirac Delta Function
Application: Lanchester's equations
Application: Electrical networks
PDEs
Separation of Variables
Heat Equation., Fourier's solution
Fourier Series
Convergence, Dirichlet's theorem
Fourier Sine Series, Fourier Cosine Series
Heat Eqn. Ends at Zero