2018-03.14-V2.ipynb

Kernel: SageMath (stable)

%display latex

Ableitungen

x = var('x')
f = x^2 * exp(2*x+3)
f

f.diff(x)

f.diff(x).collect(exp(2*x+3))

f.diff(x, x).collect(exp(2*x+3))

f.diff(x).factor()

df = f.diff(x)

Integration

ff = df.integral(x)
ff

ff.factor()

Matrizen

M = Matrix([[1,2,3], [4,5,6], [7,8,9]])
M

M.kernel()

M * vector([1,-2,1])

Gewöhnliche Differentialgleichungen

phi = function('\phi')
y = phi(x)
y

dgl = y.diff(x,x) + 4*y == 0
dgl

lsg = desolve(dgl, y)
lsg

lsg.variables()

K1, K2, dummy = lsg.variables()
y1 = lsg.subs({K1: 1, K2: 0})
y2 = lsg.subs({K1: 0, K2: 1})
y1, y2

pl1 = plot(y1, -5, 5)
pl2 = plot(y2, -5, 5, color='red')
pl1 + pl2

dgl = y.diff(x) == sqrt(x*y)
dgl

lsg = desolve(dgl, y)
lsg

lsg2 = solve(lsg, y)[0]
lsg2

glg2 = (lsg2^2).expand().canonicalize_radical()
glg2

lsg3 = solve(glg2, y)
lsg3

g = lsg3[0].rhs()
g.variables()

C = g.variables()[0]

pl1 = plot(g.subs({C:0}), 0, 2, thickness=2)
pl2 = plot(g.subs({C:-1}), 0, 2, color='red', thickness=2)

pl1 + pl2

pl2a = plot(g.subs({C:-1}), 0, (3/2)^(2/3), color='red', thickness=2, linestyle="--")
pl2b = plot(g.subs({C:-1}), (3/2)^(2/3), 2, color='red', thickness=2)
pl3 = plot(g.subs({C:1}), 0, 1.4, color='green', thickness=2)
pl1 + pl2a + pl2b + pl3