057cd029-f9b0-44ef-b2d4-c9388eeb7d24 38 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

x ProductLog e 1 exp : u x e > x ln x ln ln - x 1 e / + sqrt 1 e / sqrt - if 11 line : w 2 u * : u w - abs u abs / 1010 -^> w u : u u 2^x e u -^* + 1 u + / eval : 21 line while u 41 line 21 line :

ProductLog is the Mathematica Lambert(W). This approximation is from Mathcad 11 Mathsoft Collaboratory, does NOT plot scalar [can't parse]

L H N xd N+1 pts L...H U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 41 line :

L 1 e / - :

H e :

L 0.3679 -

N 200 :

U L H N xd :

W i 1 N 1 + range sol i el U i el ProductLog eval : for sol 21 line :

W U W augment :

x F x x exp * :

x F W 2 1 sys

With this initial Smath will output the 1rst branch of the most common Lambert [W]. You can assign for in situ calculation. Unlike ProductLog, MySol(w) can't be discretized.

init 0 :

w MySol x x exp * w - 0 ≡ x init roots :

w 0.25 - :

w MySol 0.357402956181389 -

w ProductLog eval 0.357402956181391 -

x Ω x ProductLog eval :

... 'eval(,)' to parse discrete ...

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

t C Km t Vm Km / * (- C0 C0 Km / exp * ln Km / + exp Ω * :

L H N xd N+1 pts L...H U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 41 line :

L 0 :

H 10 :

N 200 :

U L H N xd :

DE i 1 N 1 + range p i el U i el Vm Km / * (- C0 C0 Km / exp * ln Km / + exp : d i el Km p i el Ω * : 21 line for d 21 line :

DEsol U DE augment :

Title 2 1.85 DE solution in Pharmacology 10 black 15 mat :

DEsol Title 2 1 sys

t C t diff Vm t C * - Km t C + / ≡

The tricky way working with W(x)

y x x exp * ≡ x solve maple y LambertW

x α μ D μ x * α x * - exp * :

y x α μ D ≡ x solve maple α y * μ / - LambertW α / -

x W e 1 exp : u x e > x ln x ln ln - x 1 e / + sqrt 1 e / sqrt - if 11 line : w 2 u * : u w - abs u abs / 1010 -^> w u : u u 2^x e u -^* + 1 u + / eval : 21 line while u 41 line 21 line :

x L x W eval :

Plot eval(,,) Lambert W(x)

x L

α 1 :

μ 1 :

x φ α x * μ / - L α / - (:

x α μ D

1 e / φ 1

This function is complex for x >1/e size the plot within the real range

x φ