Various systematized diffeq solvers.
A solver for
(d/dx)y = ay
y(0) = y0
A solver for
(d/dx)y = x e^(x^2)
y(0) = y0
A solver for
a/y(x) + (d/dx)y(x) = 0
y(0)=y0
e.g.
x = [0,80]
a = 100
y0 = 90
[0.0, 100.0]
[10.0, 90.55385138137417]
[20.0, 80.0]
[30.0, 67.82329983125268]
[40.0, 52.91502622129181]
[50.0, 31.622776601683793]
[60.0, NaN]
[70.0, NaN]
A solver for
(d/dx)y = xcos(x)
y(0) = y0
e.g.
y(0)=200
[0.0, 200.0]
[3.0, 198.43336752757915]
[6.0, 198.28367729745682]
[9.0, 201.79793610529114]
A solver for
(1/y) (d/dx)y = x^a
y(0) = y0
e.g.
a = 0.9, y0 = 10
[0.0, 10.0]
[0.5, 13.467777004085889]
[1.0, 30.377317775174827]
[1.5, 110.3046788835147]
[2.0, 632.321766447662]
A solver for
(d/dx)^2 y(x) + y(x) = 0
y(0) = a
y'(0) = b
e.g.
a = 100 b = 10
[0.0, 100.0]
[0.5, 92.5525115750793]
[1.0, 62.44494043489294]
[1.5, 17.048670032810836]
[2.0, -32.521709386457424]
A solver for the homogenous second order ODE with Neumann boundary conditions.
ay'' + by' + cy = 0
y(0)=y0
y'(0)=yp0
Case underdamped,
a = 2, b = 1, c = 10
y(0) = 10
y'(0) = 5
r0 -0.25
s0 2.222048604328897
c1 10.0
c2 2.4751933820372525
[0.0, 10.0]
[1.0, -3.18782205013981]
[2.0, -3.0553925771642585]
[3.0, 4.818387067593098]
[4.0, -2.696290137157051]
[5.0, -0.3766856496777386]
[6.0, 1.991045969519758]
[7.0, -1.6514569955638314]
[8.0, 0.35166099240638543]
[9.0, 0.6696240322022905]
Case critically damped,
a = 1, b = 2, c = 1
y(0) = 3
y'(0) = 7
reins -1.0
ceins 3.0
czwei 10.0
[0.0, 3.0]
[1.0, 4.782432735228751]
[2.0, 3.112711514442092]
[3.0, 1.64297325613951]
[4.0, 0.7875724722155697]
[5.0, 0.35711119095152977]
[6.0, 0.1561613871299806]
[7.0, 0.06656738348547968]
[8.0, 0.027843398115908483]
[9.0, 0.0114771117800612]
Case overdamped,
a = 1, b = -2, c = -8
y(0) = 2
y'(0) = 2
reins 4.0
rzwei -2.0
ceins 1.0
czwei 1.0
[0.0, 2.0]
[1.0, 54.73348531638085]
[2.0, 2980.976302680617]
A solver for the forced second order ODE,
ay'' + by' + cy = f0 cos(kx)
e.g.
y(0) = 0
y'(0) = 0
k = 2pi
f0 = 1000
a = 1
b = pi
c = 100pi^2
[0.0, -2.2712781110811875E-4]
[1.0, 0.8399685821831873]
[2.0, 1.016404409689213]
[3.0, 1.046596965276487]
[4.0, 1.0501377429332373]
[5.0, 1.050079936344021]
[6.0, 1.0498800539909674]
[7.0, 1.0498013373930988]
[8.0, 1.0497791557937826]
[9.0, 1.0497740200403944]