<content> < Expressions > </content>

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Isolating expressions

Expression replacement `E `a `b ERep equrep(`E, `a num2str, `b num2str) concat str2num 11 line :

Expression replacement by rules `E `R ERules `A `E : `Q `T `F #EIF# `Q eval num2str 1 ≡ `T `F if 11 line : 12 mat `k 1 `R rows range `R cols 2 ≡ (`R `k 3 el 1 ≡ (| `r `R `k 2 el : `r #EIF#(`R `k 3 el, `R `k 2 el, `R `k 1 el) concat : if `A equrep(`A, `R `k 1 el, `r) concat str2num : 21 line for `A 31 line :

Expression replacement by rules n times `E `R `n ERules `A `E : `iter 1 `n range `A num2str `A `A `R ERules : num2str ≡ break if for `A 31 line :

Expression to Vector `A EVec `V 0 : `n 0 : 12 mat `a `A sys2mat.1 `V `n `n 1 + : el `a : for `V 31 line :

Expression dependency `E `x EDep `E `x `#XDep# ERep num2str `#XDep# num2str findstr num2str -1 ≠ eval 11 line :

Expression derivative `E `x EDiff `s diff(`E `x `#XD# ERep num2str,`#XD#) concat : `s str2num `#XD# `x ERep 21 line :

Num to Expression `E `O N2E `E ≡ num2str 1 ≡ `a `b #Div# `a `b #neg# #Pow# #Times# : `A `E #EHold# `a ¬ `a #bNO# equrep : `A `A `a `b > `b `a < equrep : `A `A `a `b ≥ `b `a ≤ equrep : `s `k `A equrep(`A,a `s b, #EB `k 2 el (a,b)) concat : `k 5 #EB rows range `A #EB `k 1 el `k `A str2num : for `A `A num2str - +#neg#* strrep str2num : `k 14 range `A `A num2str {(strrep}) strrep : `A `A (#N2E#(strrep #EB `k 1 el #Eb strrep str2num : `iter 1 #EN range `A num2str `A #Eb `k `A str2num : num2str ≡ break if for `A `A num2str #N2E#((strrep #Eb #EB `k 1 el strrep str2num : 41 line for `A `A #ES #EN ERules : `a #N2E# `a : `a #EHold# `a : 13 mat `O ≡ `A `a `b #b00# `a #neg# `b #Times# #Plus# : `n 0 : `B 0 : 13 mat `b # `O # strrep! strrep #(concat : `a `A sys2mat.1 `α `a num2str : `α `b concat `b findstr 1 el 1 ≡ `B `n `n 1 + : el #b00#(`α `b strlen 1 + substr concat str2num : `O! findstr 1 - ≡ `B `n `n 1 + : el `a : if 21 line for `B 41 line if 101 line if 11 line :

This avoid the bug:

a b & (c | (d ≡ num2str str2num a b & (c d ≡ (|

This is, can't use num2str with booleans

a b & (c | (d ≡ num2str a&b|c≡d

Expression to Num `E `O E2N `O ≡ `k #EB rows 1 range `δ #EB `k 2 el (a,b):(a #EB `k 1 el b) concat str2num : for #neg# 1 - : `a exp e `a^: 12 mat `E `a #bNO# `a ¬ equrep 31 line `E `O N2E E2N if 11 line :

Num to Expression `E N2E `E N2E 11 line :

Expression to Num `E E2N `E E2N 11 line :

Binary Expressions #EB + *^/ < ≤ ⨁ & | ≡ ≠ ≈ ≉ #Plus# #Times# #Pow# #Div# #bLT# #bLE# #bXO# #bAN# #bOR# #bEQ# #bDI# #bAE# #bAD# 213 mat transpose :

Replacing deep #EN 10 :

Binary symbol helper #Eb ⨁ :

#ES 02 matrix :

#ES #ES #Plus#(0,a) a 12 mat stack :

#ES #ES #Plus#(a,0) a 12 mat stack :

#ES #ES #Times#(1,a) a 12 mat stack :

#ES #ES #Times#(a,1) a 12 mat stack :

#ES #ES eval :

Isolation utility `E# `x# isol `E `x `i `y `x N2E : `a `b `E `a E2N `b E2N #bEQ# N2E 11 line : 12 mat `a `b `f `h EQ#.A `a `y EDep `a `h `E `f `b `E if : `a `b `z `f `h `j EQ#.B `a `y EDep `b `y EDep `k `f `y 0 ERep : `f `k - `z `k - `E 21 line `a `j `E if `b `y EDep `b `h `E `f `z `E if if : `A `E `x `y ERep num2str ≡ strsplit 0 stack 12 range el : `A `A str2num vectorize : `a `A 1 el numden : `b `A 2 el numden : 13 mat `A `a 1 el `b 2 el * `a 2 el `b 1 el * - : `D `A `y EDiff : 12 mat `D2 `D `y EDiff : num2str 0 ≡ (`D num2str 0 ≠ (& `A `y `A `y 0 ERep `D / - #bEQ# : `D2 num2str 0 ≠ (`D2 `y EDiff num2str 0 ≡ (& `b `D `y 0 ERep : `a 0.5 `D2 * : 12 mat `A `y #σ# `b 2^4 `a * `A `y 0 ERep * - sqrt * `b - 2 `a * / #bEQ# : 21 line `A `A N2E 0 #bEQ# #i #EN ERules : if `a `b #bEQ# `q `a num2str `y num2str ≡ (`b `y EDep ¬ (& (: (`q `b `a `b ≡ if 21 line : `A E2N 91 line : `E `E# EVec : `x `x# EVec : `N `E rows : `M 0 : `R 0 : 15 mat `N 1 > `m 1 `N 1 - (range `n `m 1 + (`N range `E `n el `E `n el `x `m el `E `m el `x `m el `i ERep : for for `m `N 1 range `E `m el `E `m el `x `m el `i : `n `N 1 - (1 range `E `n el `E `n el `x `m el `E `m el ERep : for 21 line for 21 line `E `E 1 el `x 1 el `i : if `S `E num2str : `k `σ'σ(`k num2str) concat : `k 0 : 13 mat `f `S #σ# findstr 1 - : (2 - ≠ `S `S 1 `f 1 el substr `k `k 1 + : `σ `S `f 1 el 4 + substr concat : while `S str2num 61 line :

<content> isol rules </content>

#i 02 matrix eval :

#ia sin cos tan sec csc cot sinh cosh tanh sech csch coth exp pow10 erf asin acos atan asec acsc acot asinh acosh atanh asech acsch acoth ln log10 erfinv 215 mat transpose :

k# 1 #ia rows range #i #i #bEQ#(#ia k# 1 el (a),b) concat EQ#.A(a,b, #ia k# 1 el (a), #ia k# 2 el (b) concat 12 mat stack eval : for

k# 1 #ia rows range #i #i #bEQ#(#ia k# 2 el (a),b) concat EQ#.A(a,b, #ia k# 2 el (a), #ia k# 1 el (b) concat 12 mat stack eval : for

#i #i #bEQ#(#Pow#(a,b),0) #bEQ#(a,0) 12 mat stack :

for Re(b)>0

#i #i #bEQ#(#Pow#(a,b),1) #bEQ#(a,1) 12 mat stack :

for a,b reals. If not: a!=0, log(a)!=0,
b = (2 i π n)/log(a), n element Z

#i #i #bEQ#(#Pow#(a,b),z) EQ#.B(a,b,z,a^b,ln(z)/ln(a),#σ#*(z^(1/b))) 12 mat stack :

sign

#i #i #bEQ#(#Times#(a,b),z) EQ#.B(a,b,z,a*b,z/a,z/b) 12 mat stack :

#i #i #bEQ#(#Plus#(a,b),z) EQ#.B(a,b,z,a+b,z-a,z-b) 12 mat stack :

#i #i #bEQ#(#Times#(a,b),0) cases(#bEQ#(a,0),#σ#,#bEQ#(b,0)) 12 mat stack :

#i #i eval :

#i rows 36

k# #ia Clear 1

<content> isol examples </content> <content> Examples </content>

eq expr isol

try to isolate expr in the eq.

eq c t * a + b ≡ :

eq t isol a b - c / -

eq c t * isol a b - (-

eq a b θ 2^tan * + c ≡ :

eq θ isol c a - b / atan sqrt 1 σ *

eq θ 2^isol c a - b / atan

eq a cos b t n^exp * + c ≡ :

eq t isol c a cos - b / ln n nthroot 1 σ *

eq a isol c 1 - (b * e t n^^* + acos

eq n isol c a cos - b / ln ln t ln /

eq a ln b u n^ln / + c ≡ :

eq u isol 1 e b c - a ln + (n * /^/

eq a isol e b - c n * u ln * + n u ln * /^

eq n isol b u ln c - a ln + (* / -

eq t c a cosh - b / ln n nthroot ≡ :

eq a isol c - e t n^1 σ *^b * + (- acosh

eq c isol e t n^1 σ *^b * a cosh +

eq b isol c a cosh - (1 σ * e t n^2 σ *^/

eq n isol 1 σ c a cosh - b / ln ln * t ln /

x atan sin x 1 x 2^+ sqrt /

x atan cos 11 x 2^+ sqrt /

eq ω t * φ + atan sin b ≡ :

eq ω isol ω t * b 1 φ ω t * + (2^+ sqrt - 1 φ 2^+ sqrt + (* + φ - 1 φ 2^+ sqrt b * + ≡

eq ω t * φ + atan cos b ≡ :

eq ω isol φ - b 2^- 2 σ + b 2^/ sqrt 1 σ * + t /

x x x sin * + x isol 01 σ π 2 / - cases

For complicate equations can't find a close form, and returns an equation

x 2 x * cos + 2 x * sin ≡ x isol 1 x - 2 x * sin + 2 x * cos - 1 ≡

Polynom

eq a 2^1 a 2^+ (b 2^* ≡ :

eq a isol 4 sqrt 1 b 2^- (b 2^* sqrt * 1 σ * 21 b 2^- (* /

eq a x 2^* b x * - c + 0 ≡ :

eq x isol b b 2^4 a * c * - sqrt 1 σ * + 2 a * /

eq 8 x 2^* 5 x * - 7 + 0 ≡ :

r eq x isol : 5199 - sqrt 1 σ * + 16 /

n σ 1 :

r 0.3125 0.8817 i * +

n σ 1 ± :

r 0.3125 0.8817 i * + 0.3125 0.8817 i * - 21 sys

n σ 1 - :

r 0.3125 0.8817 i * -

75 - 8 stack polyroots 0.3125 0.8817 i * + 0.3125 0.8817 i * - 21 mat

σ σ r Clear 1

x 3^2 x 2^* + 5 x * - 6 - x isol x 5 - x 2 x + (* + (* 6 ≡

x 3 - (x 2 + (* x isol 125 sqrt 1 σ * + 2 /

x a - (x b + (* x c - (2^* x isol a 1 σ b - cases 2 σ c cases

x 3^2 x 2^* + 5 x * - 6 - x isol x 5 - x 2 x + (* + (* 6 ≡

Not Systems

eq t a + cos sin b ≡ a b t 2^tan * + c ≡ 21 mat :

eq t isol vectorize a - b asin acos + c a - b / atan sqrt 1 σ * 21 mat

isolates t at each equation, not solve the system.

eq a b 21 mat isol vectorize t - b asin acos + a c - t 2^tan / - 21 mat

isolates a from the first eq, and b from the second one.

Systems

3 x * 4 y * + 3 ≡ x y - 1 ≡ 21 sys x y 21 sys isol 7 - 621 mat

x y + z - 2 ≡ x 2 y * + 3 z * - 1 ≡ 4 x * 2 y * + 3 z * - 9 ≡ 31 sys x y z 31 sys isol 211 - 31 mat

x y + s ≡ x y * p ≡ 21 sys x y 21 mat isol s - s 2^4 p * + sqrt 1 σ * + 2 / - s s 2^4 p * + sqrt 2 σ * + 2 / 21 mat

<content> Atwood </content>

m.1 m.2 g a T Clear 1

Ideal Atwood Machine.

eq m.1 g * T - m.1 a * ≡ T m.2 g * - m.2 a * ≡ 21 sys :

equation for m.1

T a 21 mat eq T a 21 sys isol :

equation for m.2

T 2 m.1 * g * m.2 * m.1 m.2 + /

a g m.1 m.2 - (* m.1 m.2 + /

Numerical
example

m.1 4 kg * : m.2 12 kg * : g g.e : 31 sys

T kgf 6

a g.e 0.5 -

Atwood Machine with pulley with inertia and friction.

m.1 m.2 m g a T.1 T.2 r α I τ.f Clear 1

eq m.1 g * T.1 - m.1 a * ≡ T.2 m.2 g * - m.2 a * ≡ T.1 T.2 - (r * τ.f - I α * ≡ a α r * ≡ 41 sys :

T.1 T.2 α 31 mat eq 13 range el T.1 T.2 α 31 sys isol :

Not solve for a yet for more compact expressions

eq for pulley

Non-sleep cond

T.1 m.1 g a - (*

T.2 m.2 g a + (*

α τ.f - m.1 g a - (* m.2 g a + (* - (r * + I /

a eq 4 el a isol : τ.f - g m.1 m.2 - (* r * + (r * I m.1 m.2 + (r 2^* + /

Sure?

τ.f 0 : I 0 : 21 sys

we recover the
ideal case

T.1 2 m.1 * m.2 * g * m.1 m.2 + /

T.2 2 m.2 * m.1 * g * m.1 m.2 + /

a g m.1 m.2 - (* m.1 m.2 + /

Numerical
example

m.1 4 kg * : m.2 12 kg * : g g.e : 31 sys

Pulley data r 12 cm * : M 0.5 kg * : I 0.5 M * r 2^* : τ.f 0.1 N * m * : 41 sys

Pulley radious

T.1 kgf 5.9901

mass

T.2 kgf 6.0296

Inertia

a g.e 0.4975 -

frictional torque

Almost the same values for the ideal case, for τ.f small.

<content> Capacitor </content>

Parallel Capacitors

Q q.1 q.2 V C.1 C.2 C.eq Clear 1

q.1 q.2 V C.eq 41 mat Q q.1 q.2 + ≡ q.1 C.1 V * ≡ q.2 C.2 V * ≡ Q V C.eq * ≡ 41 sys q.1 q.2 V C.eq 41 mat isol :

q.1 Q C.1 * C.1 C.2 + /

q.2 Q C.2 * C.1 C.2 + /

V Q C.1 C.2 + /

C.eq C.1 C.2 +

Q V.1 V.2 V C.1 C.2 C.eq Clear 1

Sereies Capacitors

Q V.1 V.2 C.eq 41 mat Q V.1 C.1 * ≡ Q V.2 C.2 * ≡ V V.1 V.2 + ≡ Q V C.eq * ≡ 41 sys Q V.1 V.2 C.eq 41 mat isol :

V.1 V C.2 * C.1 C.2 + /

V.2 V C.1 * C.1 C.2 + /

Q V C.2 * C.1 * C.1 C.2 + /

C.eq C.1 C.2 * C.1 C.2 + /

<content> Example </content>

Expression Isolation

α β ω.1 ω.2 D Clear 1

Equations

D 0.5 α ω.1 / β ω.1 * + (* ≡

eq α β 21 sys isol 2 D * ω.1 * ω.2 * ω.2 ω.1 + / 2 D * ω.2 ω.1 + / 21 mat

D 0.5 α ω.2 / β ω.2 * + (* ≡

Quasi-Symbolic
solutions

α eq 1 el α isol :

β eq 2 el β isol :

β 2 D * ω.2 ω.1 + /

α 2 ω.1 * D * ω.2 * ω.2 ω.1 + /

Numerical values

D 0.03 :

ω.1 20.33 :

ω.2 61.1 :

Numerical solution

α 0.9153

β 0.0007

α β ω.1 ω.2 D Clear 1

<content> Example </content>

Expression Isolation

Two radicals is too much.

https://smath.com/en-US/forum/topic/59Zah-last/How-to-solve-these-two-equations

s.1 k m / sqrt 4 ≡ k m 2 + / 9 ≡ 21 sys m k 21 sys isol (: 181 σ * 7 / 2887 / 21 mat

s.2 k m 2 + / sqrt 3 ≡ k m / 16 ≡ 21 sys m k 21 sys isol (: 27 - 161 σ * + (* 7 / 2887 / 21 mat

Taking

x σ 1 :

s.1 187 / 2887 / 21 mat

s.2 187 / 2887 / 21 mat

x σ Clear 1

<content> Example </content>

https://smath.com/en-US/forum/topic/ysQFE/How-to-get-real-roort-in-symbolic_

R2 R.a 2^≡

β2 β 2^≡

then

β β2 2^1 β2 - (2^2 ζ * (2^β2 * + / β2 diff β2 isol sqrt : 2 - 42 nthroot 1 σ * + sqrt 4 sqrt 12 ζ 2^* - (- sqrt * /

n σ 1 - :

β simplify maple 112 ζ 2^* - sqrt /

β n σ Clear 1

Alvaro

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