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}

\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


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

Heat Eqn. Both Ends Insulated