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-(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

2nd Translation Theorem

Derivative thrms, Solving DEs

Convolution theorem

You can find the convolution of any piecewise defined function with another (off the domain of definition, they are assumed to be zero). Here is f , f*f , and f*f*f , where f(x)=1 , 0<x<1 :

sage: x = PolynomialRing(QQ, 'x').gen()
sage: f = Piecewise([[(0,1),1*x^0]])
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))

The command show(P+Q+R) displays this:

$http://sage.math.washington.edu/home/wdj/art/convolutions.png$

Though SAGE doesn't simplify very well, you can see that the LT(f*f) is equal to LT(f)^2:

sage: f.laplace(x,s)
1/s - e^(-s)/s
sage: g.laplace(x,s)
-(s + 1)*e^(-s)/s^2 + (s - 1)*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2
sage: (f.laplace(x,s)^2).expand()
-2*e^(-s)/s^2 + e^(-(2*s))/s^2 + 1/s^2

Dirac Delta Function

Application: Lanchester's equations

Application: Electrical networks

PDEs

Separation of Variables

Heat Equation., Fourier's solution

Fourier Series

If f(x) is a piecewise-defined polynomial function on -L<x<L then the Fourier series

\displaystyle f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],

converges. In addition to computing the coefficients a_n,b_n , it will also compute the partial sums (as a string), plot the partial sums (as a function of x over (-L,L) , for comparison with the plot of f(x) itself), compute the value of the FS at a point, and similar computations for the cosine series (if f(x) is even) and the sine series (if f(x) is odd). Also, it will plot the partial F.S. Cesaro mean sums (a smoother" partial sum illustrating how the Gibbs phenomenon is mollified).

sage: f1 = lambda x:-1
sage: f2 = lambda x:2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_cosine_coefficient(5,pi)
-3/(5*pi)
sage: f.fourier_series_sine_coefficient(2,pi)
-3/pi
sage: f.fourier_series_partial_sum(3,pi)
-3*sin(2*x)/pi + sin(x)/pi - 3*cos(x)/pi + 1/4

Plotting the partial sums is implemented: Typing f.plot_fourier_series_partial_sum(15,pi,-5,5) yields

$http://sage.math.washington.edu/home/wdj/art/fourier-partial-sum1.png$

and typing f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5) yields the much smoother version:

$http://sage.math.washington.edu/home/wdj/art/fourier-partial-sum2.png$

Differential_Equations