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Laplace DE solvers from Mathcad 11 => Smath

These Laplace solvers are sourced in advanced Engineering books, adapted around Mathcad 11/ Maple companions. 1. Sokolnikoff & Redheffer [Berkeley, UC]. 2. Mercier [Arts & Métiers, Angers]. 3. Institut Professionnel [Paris]. __________________________________ In this Smath document, we will attempt to survive these solvers from Mathcad. Each solver is given under a forcing function q(t), c/w an example, sanity checked Mathcad 11. __________________________________ Besides the Laplace solvers, we have learned: 1. Setup the local definition. 2. Introduce the silent 't' that makes the local definition work, i.e: the solvers work.

1. 1rst Order Linear Differential Equation

1. Laplace of the forcing function q(t)

t q t 1 + (e 3 t * -^* :

s L t q t s laplace maple :

s L 3 s + 3 s + (2^+ 3 s + (3^/

2. Dress the solver for Laplace

Y0 - s L - m s * n s / + / (- Y0 s L + (s * m s 2^* n + / ≡

<= Recast the model from source, Maple likes "expression" F(s) in Laplace work.

s F Y0 s L + (s * m s 2^* n + / :

3. Instruct Smath wisely with the local definition

Y0 m n t fnct't' t 11 line : s F s't' invlaplace maple 21 line :

<= A most gorgeous instruction ! The blue comes by default in typing t' .... then back 't'

4. Export the solution wrt the DE parameters

Y0 m n t fnct n m / sqrt t * cos n 219 Y0 * + (* m * Y0 n * + (* 9 m 2^* 49 Y0 * + (* + (* - m n m / sqrt n m / sqrt t * sin * n 15 m * + (* - 23 t * - exp * n * + 363 t * - exp * m * + (* + 3 t * 3 t * - exp * n 9 m * + (* m * + n 9 m * + (2^m * / -

5. Assign parameters to this "forced DE"

m 1 :

n 16 - :

Y0 0 :

t f Y0 m n t fnct :

6. The executable solution from symbolic expansion f(t)

t f 44 t * exp 3 t * - exp - (* 21 t * 3 t * - exp * + 49 /

Laplace 1rst order, forced solution

x 0 ≥ ( x 1.25 ≤ ( & x f Outside reasonable DE solution if

x sol 44 x * exp 3 x * - exp - (* 21 x * 3 x * - exp * + 49 / :

Note: X_Y plot refuses to understand something in there ! Don't attempt to plot sol(x): it crashes Smath. Maybe too much for X_Y plot ?

The forcing function q(t), Laplace (s)

x 0 ≥ ( x 1.25 ≤ ( & x q Undefined if x 0 ≥ ( x 1.25 ≤ ( & x L Undefined if 2 1 sys

Working more Maple/Smath

This suite is not related to DE solvers, just something interesting, Edu. Observe: invlaplace, s, t laplace, t, s

μ 5 :

T μ s d 1 T s * 1 + (μ^/ :

t XFR T μ s d s t invlaplace maple : t 4^t T / - exp * 24 T 5^* /

# t n 1 - (^n 1 - (! / t - T / exp T n^/ * ≡

EQ.0

n 4 :

T 1 :

t xfr 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

# 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - ≡

EQ.1

t f t xfr t int :

t f maple t 43214 t - exp * + (* 3 t - exp * t * + (* 2 t - exp * t 2^* + (* t - exp t 3^* + (* 120 t - exp * + 24 /

t Int t 43214 t - exp * + (* 3 t - exp * t * + (* 2 t - exp * t 2^* + (* t - exp t 3^* + (* 120 t - exp * + 24 / :

x 0 ≥ ( x 20 ≤ ( & x Int 0 Int - Cumulative integral XFR if x 0 ≥ ( x 20 ≤ ( & x xfr XFR multiconstants if

The Maple Laplace is reversible. You can rename T = Tc but can reintroduce 'n'.

n 5 :

s Laplace t n 1 - (^n 1 - (! / t - Tc / exp Tc n^/ * t s laplace maple :

s Laplace 11 s Tc * + (5^/

π sqrt 2 s 32 /^* / t 0 > assume s t invlaplace maple t sqrt

π sqrt 2 s 35 /^* / t 0 > assume s t invlaplace maple π sqrt 2 t 25 /^* 35 / GAMMA * /

Not wise enough to answer !

35 / Gamma 1.4892

π sqrt s 32 /^/ t 0 > assume s t invlaplace maple 2 t sqrt *

π sqrt 0.5 s 32 /^(* / t 0 > assume s t invlaplace maple 4 t sqrt *

n n Clear :

m m Clear :

Y0 Y0 Clear :

2. 2nd Order Homogeneous Differential Equation

Homogeneous 2nd order: Y"(t) + m*Y'(t) + n*Y(t)= q(t

t ω d q d t * - exp ω * :

s ω d L t ω d q t s laplace maple :

s ω d L ω s d + /

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

Y0 Y1 m n ω t fnct't' t 11 line : s F s't' invlaplace maple 21 line :

Y0 1 :

Y1 0.5 :

m 0.5 :

n 1.5 :

d 0.5 :

ω 2 :

t f Y0 Y1 m n ω t fnct :

=> Smath bug in there: symbolic expansion is corrupted <=

There is "shadow in the context", maybe caused by some undefinable definitions. The correct expansion was done in blank work sheet. This type of inconvenience is not common but it occurs and may result from some persisting code from the construction stage that can't be removed and stay undetected. Turn PC OFF and restart Smath may help. Heavy maths don't scratch Smath, but lenghtly creation DOES torture Smath as well as any CAS.

t f 23 sqrt 23 sqrt 4 t 2 / - exp * t 4 / - exp t 23 sqrt * 4 / cos * - (* 13 t 4 / - exp * t 23 sqrt * 4 / sin * + (* 69 / :

t sol 0.33333 1.19896 t * cos * - 0.903565 1.1989 t * sin * + (0.25 t * - exp * 1.33333 0.5 t * - exp * + :

x 0 ≥ ( x 10 ≤ ( & x f Outside reasonable DE solution if x 0 ≥ ( x 10 ≤ ( & x sol Outside reasonable DE solution if

Sanity check

t sol Mathcad 11 sanity check ≡

1.23456789 f 1.35795

1.23456789 sol 1.35795

jmGiraud: 2015/10/14