<content> <p> Export Specific User Parameters ... i.e plotting function </p> </content>

b6627b4e-bd9a-4b65-a19d-f76b58e7473f 43 4 5 false false 0 0 decimal

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

ellipses

https://mathcurve.com/courbes2d/pendule/pendule.shtml

k 1 - ≥

'a' sets the span ... 'k' sets separation b 0.75 : a 1 : k 1 : 13 mat

x plot 0.75 k x a / cos + sqrt * ± 1 k x a / cos + sqrt * ± 1.25 k x a / cos + sqrt * ± π 0 + 40 black 15 mat 20 x * sin 0.75 * k x a / cos + sqrt * 51 sys :

x b y b k x a / cos + sqrt * ± :

x 0.75 y x 1 y x 1.125 y x 1.25 y 4 1 sys x plot

https://math24.net/nonlinear-pendulum.html

Pendulum Homogeneous DE ...

2nd Order Type 3

UnitStep 1 ≡ UnitLinear t ≡ UnitGrow d t * exp ω * ≡ UnitDecay d t * - exp ω * ≡ Fails Dirac Integral t sqrt ≡ 51 line

t sqrt t s laplace maple π sqrt 2 s 2 nthroot 3^* /

final rest position datum 0 :

t ω d q datum :

s ω d L t ω d q t s laplace maple :

This Smath document reproduces
1/1 the original Mathcad 11.
Seeminly, unique ... NO other CAS

'ω' wild Laplace suite s ω d L 0

s F Y0 s - m - (* Y1 - s ω d L - s 2^s m * + n + (/ (- :

Laplace solver homogeneous ODE
c/w forcing function, parameters
and initial conditions.

solution is wild wrt parameters and assigned to a function wrt parameters. Y0 Y1 m n ω t fnct s F t 0 > assume s t invlaplace maple 11 line :

Export the solution wrt the DE parameters  ... a long expansion that will solve for any [Y0,Y1, m,n] Y0 Y1 m n ω t fnct m t * 2 / - exp Y0 4 n * m 2^- sqrt t * 2 / cos * 4 n * m 2^- sqrt * 2 Y1 * 4 n * m 2^- sqrt t * 2 / sin * + Y0 m * 4 n * m 2^- sqrt t * 2 / sin * + (* 4 n * m 2^- sqrt /

Y0 1 :

Y1 0 :

m 0.5 :

n 1.5 :

<content> <p> Export Specific User Parameters ... i.e plotting function </p> </content>

The RHS algorithm is the solutionwrt user parameters, assigned f(t).OBSERVE: ►  ω is wild ... from L(s,ω,d) t f Y0 Y1 m n ω t fnct :

Export this elegant user solution ... Optimiz: NONE ... copy/paste the RHS t f t 4 / - exp 2312 /^t * 4 / cos * t 4 / - exp 2312 /^* 2312 /^t * 4 / sin * 23 / +

x λ 10 x ≤ (x 20 ≤ (& cases :

x f x λ * x f x diff x λ * 2 1 sys

100 f 0

XY x 020 0.1 range : j 1 x rows range xy j el x j el f eval : for x xy augment 31 line :

τ 1100 range :

t X XY 1 col t el : t Y XY 2 col t el : 12 mat

t 1 ≡ path t X t Y 1 2 mat : path path t X t Y 1 2 mat stack : if XY 2 1 sys

'a' sets the span ... 'k' sets separation b 0.75 : a 1 : k 1 : 13 mat

x b y b k x a / cos + sqrt * ± :

x 0.75 y x 1 y x 1.125 y x 1.25 y π 0 + 40 black 1 5 mat 5 1 sys

L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L π - : H π : N 100 : 13 mat

u L H N xd :

x f b k x a / cos + sqrt * (- :

P u u f vectorize augment :

pnd P P 1 col reverse P 2 col reverse augment stack :

τ 1200 range :

t X pnd 1 col t el : t Y pnd 2 col t el : 12 mat

bob t X t Y . 40 black 15 mat :

L 2 :

track 0 L 0 L t X t Y 32 mat :

t 1 ≡ path t X t Y 1 2 mat : path path t X t Y 1 2 mat stack : if P bob track 7 1 sys

200 X i 2 * + arg ° 147.5184

200 X 3.14159265358979 -

π 3.14159265358979

Max elongation π 2^22^+ sqrt 3.7242

Min elongation 50 Y 2 - abs 3.0601

data P :

Finite differences derivative X data 1 col : Y data 2 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : Arc integrand .................... xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 181 line

Derivative fnct

D data 2 1 sys

Arc integrand

Arc

Cumulative arc length

'a' sets the span ... 'k' sets separation"'b' sets amplitude b 1.25 : a 1.19169803067892 : k 0.875 : 13 mat

mutually reactive

x b y b k x a / cos + sqrt * ± :

x 0.75 y x 1 y x 1.125 y x 1.25 y π 0 + 40 black 1 5 mat 5 1 sys x b y π 0 + 40 black 1 5 mat 2 1 sys

jmG: 20230203