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Advanced Integration . . . multi use

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3 appVersion 0.98.6179 ≡

Example: 1

solving wrt 'y' a k u t f u y a y - (* / y int k t int ≡ y solve maple :

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

solving wrt 't' a k u t f u y a y - (* / y int k t int ≡ t solve maple :

a k u t f u y a y - / ln * k a * /

a k u x f1 u a 1 k x * a * u / exp + / ln * k x * a * + u / exp :

a k u x f2 u x a x - / ln * k a * / :

a 1 : k 1 : u 1 : 13 mat

4 a k u 10 x * f1 * 1 a k u 1 x * f2 * 2 1 sys

4 a k u 0 f1 * a k u 0.5 f2 21 mat 2021 mat

a k u t Clear 1

Example: 2

u 1 u 2^+ / u 1 y int u sin - u 0 t int ≡ y solve maple 21 t cos - (* exp 2 - 21 t cos - (* exp + (* - sqrt 21 t cos - (* exp / 21 t cos - (* exp 2 - 21 t cos - (* exp + (* - sqrt 21 t cos - (* exp / - 21 mat

u 1 u 2^+ / u 1 y int u sin - u 0 t int ≡ t solve maple 1 y 2^+ ln 2 / 2 ln 2 / - 1 + acos

t Φ 21 t cos - (* exp 2 - 21 t cos - (* exp + (* - sqrt 21 t cos - (* exp / ± 1 t 2^+ ln 2 / 2 ln 2 / - 1 + acos 21 sys :

x Φ 1 el x Φ 2 el 2 1 sys

Example: 3

u u 1 u - (* / u y0 y int ρ u t0 t int ≡ y solve maple 1 ρ t - t0 + (* exp 1 - y0 + (* +

u u a u - (* / u y0 y int ρ u 0 t int ≡ y solve maple a 1 ρ t * - exp - (* ρ t * - exp y0 * +

1 x n^/ x C0 C1 int 1 y 0 t int ≡ C1 solve maple C0 ln n * 1 C0 - t C0 ln n * exp * 1 - n + (* + / - ln + 1 - n + / exp

1 x n^/ x C0 C1 int k - (1 y 0 t int * ≡ k solve maple C0 1 n -^1 C1 1 n -^C0 1 - n +^* - (* t 1 - n + (* / -

x 2^x y (2^+ 4 - (x diff x y x diff solve maple x x y * 1 - (x diff x y x diff solve maple 21 sys x x y / - x y x / - 21 sys

5 x y 2^* x y sin + x 2^- (x diff x y x diff solve maple 7 x y 4^* x 3^x y * + x + 4 - (x diff x y x diff solve maple x 2^x y 2^- 1 - (x diff x y x diff solve maple 31 sys 2 x * 10 x y * x y cos + / 13 x 2^* x y * + 28 x y 3^* x 3^+ / - x x y / 31 sys

x y Sol u 1 - 1 u 2^+ / u 0 y int 1 u + (u 0 x int + 1 u + (u 0 u int - maple :

12 / 1 y 2^+ ln * y atan - x + x 2^2 / + 4 -

x y Sol 12 / 1 y 2^+ ln * y atan - x + x 2^2 / + 4 - :

x 2 x + (* - u 2^+ 2 y atan * + 1 y 2^+ ln - 2 u * + 2 / -

Sol

x y Sol x solve maple 1 - 92 y atan * + 1 y 2^+ ln - sqrt + 192 y atan * + 1 y 2^+ ln - sqrt + (- 21 mat

x plot 1 - 92 x atan * + 1 x 2^+ ln - sqrt + 192 x atan * + 1 x 2^+ ln - sqrt + (- 21 mat :

x plot 1 el x plot 2 el 2 1 sys

Example: 4

v Vg * v 2^2 / - (1 -^v V0 Vt int K 1 t 0 T int * ≡ Vt solve maple 2 V0 * Vg * K T * Vg * exp * V0 1 - K T * Vg * exp + (* 2 Vg * + /

v t diff K v Vg * v 2^2 / - (* ≡

K Vg V0 t q 2 V0 * Vg * K t * Vg * exp * V0 1 - K t * Vg * exp + (* 2 Vg * + / :

x c 10 x ≤ cases :

10 5 1 x q x c * 10 - 5 1 x q x c * 2 1 sys

a 10 : b 4 : c 0.25 : 13 mat

Verhulst a b c x q x c * 0.125 a b c x q x diff (* x c * 21 sys :

X a b c x q x 2 diff 0 ≡ x 01 solve : 0.086

Verhulst X 15 X 5 - 2 2 mat 2 1 sys

Example: 5

DE Verhulst [Logistic]

a k t Clear 1

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 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 + (* + /

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 + (* + /

a k y0 t0 t f3 1 u a u - (* / u y0 y int k u t0 t int ≡ y solve maple :

a k y0 t0 t f3 y0 a * k a * t t0 - (* exp * a y0 1 - k a * t t0 - (* exp + (* + /

yo r t to - (* exp * 1 yo - yo r t to - (* exp * + / 111 yo / 1 - (r t to - (* - exp * + / ≡

r to yo t Logistic 111 yo / 1 - (r t to - (* - exp * + / :

NOT from Smath

modelling hysterisis r 1 : yo 0.5 : to 10 : ε 0.75 : 14 mat

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

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

r to yo x Logistic r to ε - yo x Logistic r to ε + yo x Logistic to yo + 15 black 1 5 mat 4 1 sys 1.25 0.5 x f 1.25 0.75 x f 1.25 1 x f 3 1 sys 1.25 0.5 0.025 x f1 1.25 0.75 0.025 x f1 1.25 1 0.025 x f1 3 1 sys

Example: 6

Random example...

u C x Clear 1

SYMBOLIC u 1 + (- 1 u 2^+ / u int 1 x / x int C + ≡ maple 1 u 2^+ ln 2 u atan * + 2 / - C x ln + =

x f 1 x 2^+ ln 2 x atan * + 2 / - : x q 1 x 2^+ ln 2 x atan * + 2 / : 21 sys

X x f x q - 0 ≡ x 4 - 3 - solve : 3.503 -

x f x q X 1 X 1 - 2 2 mat 3 1 sys

Example: 7

Theoretical free fall

... SYMBOLIC ...
parabolic free fall m g * m g * k v * - / v int g 1 t int * - Vo + ≡ v solve maple m g * k Vo - g t * + (* m g * / exp - k /

m g k Vo t freefall m g * k Vo - g t * + (* m g * / exp - k / :

m 1 : g 9.81 : k 0.9071 : Vo 1 : 14 mat

u 0 x ≤ :

X m g k Vo x freefall 0 ≡ x 05 solve : 2.619

m g k Vo x freefall u / 0 g + 40 red 1 5 mat X 0 + 40 red 1 5 mat 3 1 sys

m g k Vo 0 freefall 9.81

m adjust gravity g [m/s²] ≡ k drag coefficient ≡ Vo initial velocity [m/s] ≡ 31 line