62208fea-0e60-4e31-bb82-7887c5f3945e 3 4 5 decimal

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Solving Verhulst

a k t f 1 y a y - (* / y int k t int ≡ y solve maple :

a k t f a k t * a * exp * 1 k t * a * exp + /

a k y0 t f1 1 u a u - (* / u y0 y int k u 0 t int ≡ y solve maple :

a k y0 t f1 y0 a * k t * a * exp * a y0 1 - k t * a * exp + (* + /

k y0 t0 t f2 1 u 1 u - (* / u y0 y int k u t0 t int ≡ y solve maple :

k y0 t0 t f2 y0 k t t0 - (* exp * 1 y0 1 - k t t0 - (* exp + (* + /

1rst Order Type 1

natural definition s F t 0 > assume s t invlaplace maple ≡

Y0 - s L - m s * n + / (- Y0 s L + (s * m s * n + / ≡

UnitStep 1 ≡ UnitLinear t ≡ UnitGrow d t * exp ω * ≡ UnitDecay d t * - exp ω * ≡ UnitSqrt t sqrt ≡ 51 line

d t * exp ω * 1 -

1. Laplace of the forcing function q(t) t q d t * - exp ω * :

s L t q t s laplace maple :

s L ω s d + /

2. Dress the solver for Laplace s F Y0 s L + (s * m s * n + / :

3. Instruct Smath wisely with the natural definition Y0 m n t fnct s F t 0 > assume s t invlaplace maple :

4. Export the solution wrt the DE parameters Y0 m n t fnct d ω d t * - exp * m * n n t * m / - exp * Y0 * - (* n n t * m / - exp * ω * - (m * n 2^n t * m / - exp * Y0 * + m 2^n - d m * + (* /

5. Assign parameters to this "forced DE"

Y0 0 :

m 1 :

n 0.25 - :

d 1 :

ω 3 :

t f Y0 m n t fnct :

6. The executable solution from symbolic expansion f(t) t f 34 t - exp * t 4 / exp + (* 5 /

Laplace 1rst order, forced solution

x 0 ≥ ( x e 2 ^ ≤ ( & x f wild domain if 1 1 sys

1. exp(-d*x)*ω generalised decay 2. exp(d*x)*ω generalised grow 3. ω=1 ... UnitGrow/UnitDecay 4. 'd' sets the shape

d x * - exp ω * ω 1 d x * - exp - ( * d x * exp ω * 3 1 sys

ω 1 d t * - exp - (* NOT Laplace s ≡