Differences between revisions 1 and 12 (spanning 11 versions)
 ⇤ ← Revision 1 as of 2007-06-24 23:27:22 → Size: 28 Editor: DavidJoyner Comment: ← Revision 12 as of 2007-06-24 23:58:15 → ⇥ Size: 2118 Editor: DavidJoyner Comment: Deletions are marked like this. Additions are marked like this. Line 2: Line 2: == 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-(x-1)^2sage: h(x) = -xsage: 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-(x-1)^3sage: 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 ====== Heat Eqn. Both Ends Insulated ===

# Differential Equations

## Laplace Transform (LT) methods

### 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-(x-1)^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-(x-1)^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

## PDEs

### Heat Eqn. Both Ends Insulated

Differential_Equations (last edited 2008-11-14 13:42:08 by localhost)