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However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented: {{{ sage: f(x) = x^2+1 sage: g(x) = 1(x1)^3 sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: P.laplace(x,s) (s^3  6)*e^(s)/s^4  ((2*s^2 + 2*s + 2)*e^(s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(3*s)/s^4 + (3*s  1)*e^(3*s)/s^2 + (5*s + 1)*e^(5*s)/s^2 + (s^2 + 2)/s^3 }}} 
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
x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} )
on (0, 1), x \ {\mapsto}\ 1  ( x  1 )^2
on (1, 3), x \ {\mapsto}\ x
on (3, 5), as follows: sage: f(x) = sin(x*pi/2) sage: g(x) = 1(x1)^2 sage: h(x) = x sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: latex(P)
However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented:
sage: f(x) = x^2+1 sage: g(x) = 1(x1)^3 sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: P.laplace(x,s) (s^3  6)*e^(s)/s^4  ((2*s^2 + 2*s + 2)*e^(s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(3*s)/s^4 + (3*s  1)*e^(3*s)/s^2 + (5*s + 1)*e^(5*s)/s^2 + (s^2 + 2)/s^3
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