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How to integrate this ode and how to sub...

C_R 1197 Maple
integration ode differential-equations
June 03 2023
2 6

The example below is an attempt to solve a boundary value problem (BVP) without manually replacing terms in expressions and some questions where there might be room for improvement. (Only the first integration step is shown to keep it short)

My objective is to have clean Maple input with minimal manual interventions or low-level manipulations (such as using op and similar commands).

NULL

> 

ode := diff(phi(s), s, s)+K*cos(phi(s)) = 0

diff(diff(phi(s), s), s)+K*cos(phi(s)) = 0

 (1)

NULL

> 

map(int, ode, s)

int(diff(diff(phi(s), s), s)+K*cos(phi(s)), s) = 0

 (2)
> 

Student[ODEs]:-Integrate(ode, phi(s))

Error, (in assuming) when calling 'Student:-ODEs:-Integrate'. Received: 'The given ODE, diff(diff(phi(s),s),s)+K*cos(phi(s)) = 0, is not exact'

  

...not a clue for me


Since I know the result: Multiplying by diff(phi(x), x) to make integration work.  Q1: Are there other ways to integrate a single step without a priori knowledge?

> 

(diff(phi(s), s))*ode

(diff(phi(s), s))*(diff(diff(phi(s), s), s)+K*cos(phi(s))) = 0

 (3)

NULL

> 

int(lhs((diff(phi(s), s))*(diff(diff(phi(s), s), s)+K*cos(phi(s))) = 0), s)+c__1 = 0

(1/2)*(diff(phi(s), s))^2+K*sin(phi(s))+c__1 = 0

 (4)

Boundary condition for s = L 

> 

Eval(diff(phi(s), s), s = L) = `φ'`[0]

Eval(diff(phi(s), s), s = L) = `φ'`[0]

 (5)

(4) evaluated at the boundary

> 

expand(map(Eval, (1/2)*(diff(phi(s), s))^2+K*sin(phi(s))+c__1 = 0, s = L))

Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)

 (6)

Q2: How to simplify expression (6) with (5) using Maple commands to get this expression for the integration constant

``

> 

c__1 = -(1/2)*`φ'`[0]^2-K*sin(Eval(phi(s), s = L))

c__1 = -(1/2)*`φ'`[0]^2-K*sin(Eval(phi(s), s = L))

 (7)

Some attempts

> 

algsubs(Eval(diff(phi(s), s), s = L) = `φ'`[0], Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)); applyrule(Eval(diff(phi(s), s), s = L) = `φ'`[0], Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)); simplify(Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L), {Eval(diff(phi(s), s), s = L) = `φ'`[0]}); Physics:-Substitute(Eval(diff(phi(s), s), s = L) = `φ'`[0], Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L))

Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)

  

Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)

  

Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)

  

Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)

 (8)

``

> 

map(`^`, Eval(diff(phi(s), s), s = L) = `φ'`[0], 2)

(Eval(diff(phi(s), s), s = L))^2 = `φ'`[0]^2

 (9)

Doing it manually

isolate(subs({Eval(0, s = L) = 0, Eval((1/2)*(diff(phi(s), s))^2, {s = L}) = (1/2)*`φ'`[0]^2}, Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L)), c__1)

c__1 = -(1/2)*`φ'`[0]^2-(Eval(K*sin(phi(s)), {s = L}))

 (10)

or

isolate(algsubs(value(Eval(diff(phi(s), s), s = L) = `φ'`[0]), value(Eval((1/2)*(diff(phi(s), s))^2, {s = L})+Eval(K*sin(phi(s)), {s = L})+c__1 = Eval(0, s = L))), c__1)

c__1 = -(1/2)*`φ'`[0]^2-K*sin(phi(L))

 (11)

subs(phi(L) = Eval(phi(s), s = L), c__1 = -(1/2)*`φ'`[0]^2-(Eval(K*sin(phi(s)), {s = L})))

c__1 = -(1/2)*`φ'`[0]^2-(Eval(K*sin(phi(s)), {s = L}))

 (12)

Q3: Are there ways to prevent commands evaluating Eval(phi(s), s = L)to phi(L)which might suggest incorrectly a new independent variable?

Q4: In (6), why is s = L sometimes in brackets {} ?

``
Download BVP_with_questions.mw

Why is the output of an inert integral n...

C_R 1197 Maple
integral inert 2d-output
June 02 2023
1 2

I am updating older files.

Executing with the attached code snippet from a worksheet created with Maple 16 outputs for an inert integral

instead of

Why is that and how can I fix my old worksheets to make them work with Maple 2023?

Maybe related: Execution with (or return) does not evaluate the document blocks. When all document blocks are expanded with "right click show command" the cursor does not advance to the next execution group. I can't remenber if this behaviour was allways like this or has changed.

A suggestion for Maplesoft: I have stopped using document blocks with hidden code since hiding and showing commands requires too much time (to many clicks and mouse movements). A simple double click on the marker of the document block could facilitate hiding and showing code allot.

Output_of_Int_not_in_2d.mw

Why do I get this sequence returned from...

C_R 1197 Maple 2023
alias
May 25 2023
1 3 
alias(b = JacobiCN(sqrt(2)*sqrt(x), sqrt(2)*_Z/2)^2);
                          lessthan, b

I could not find an explanation on the help page.

I would have expected simply b as the return value.

Update:
A worksheet that generates the output


 

RootOf(JacobiCN(sqrt(2)*sqrt(x), (1/2)*sqrt(2)*_Z)^2*_Z^2+_Z^2-2)

RootOf(JacobiCN(2^(1/2)*x^(1/2), (1/2)*2^(1/2)*_Z)^2*_Z^2+_Z^2-2)

 (1)

plot(RootOf(JacobiCN(2^(1/2)*x^(1/2), (1/2)*2^(1/2)*_Z)^2*_Z^2+_Z^2-2), x = 0 .. 5)

  

convert(JacobiCN(sqrt(2)*sqrt(x), (1/2)*sqrt(2)*_Z)^2, Elliptic_related)

1-JacobiSN(2^(1/2)*x^(1/2), (1/2)*2^(1/2)*_Z)^2

 (2)

convert(RootOf(JacobiCN(2^(1/2)*x^(1/2), (1/2)*2^(1/2)*_Z)^2*_Z^2+_Z^2-2), Elliptic_related)

RootOf(JacobiSN(2^(1/2)*x^(1/2), (1/2)*2^(1/2)*_Z)^2*_Z^2-2*_Z^2+2)

 (3)

alias(b = JacobiSN(sqrt(2)*sqrt(x), (1/2)*sqrt(2)*_Z))

lessthan, b

 (4)

``

Download alias_with_lessthan_output.mw

Question about the Maplesim version that...

C_R 1197 MapleSim
May 24 2023
1 1

Is there a way to determine the version of MapleSim used to create a model from the model file or within MapleSim when an older model was loaded?

Why can't isolate not isolate theta(q) b...

C_R 1197 Maple
isolate
May 16 2023
3 1

[6600.0*theta(q)-17000.0*theta(q-1)+14400.0*theta(q-2)-4000.0*theta(q-3) = v(q)-.20*theta(q)+.20*theta(q-1)]

[6600.0*theta(q)-17000.0*theta(q-1)+14400.0*theta(q-2)-4000.0*theta(q-3) = v(q)-.20*theta(q)+.20*theta(q-1)]

 (1)

isolate([6600.0*theta(q)-17000.0*theta(q-1)+14400.0*theta(q-2)-4000.0*theta(q-3) = v(q)-.20*theta(q)+.20*theta(q-1)], theta(q))

Error, (in isolate) invalid arguments for isolate

  

solve([6600.0*theta(q)-17000.0*theta(q-1)+14400.0*theta(q-2)-4000.0*theta(q-3) = v(q)-.20*theta(q)+.20*theta(q-1)], theta(q))[]

theta(q) = 0.1515105603e-3*v(q)+2.575709827*theta(q-1)-2.181752068*theta(q-2)+.6060422411*theta(q-3)

 (2)

It works for h(x)  in help(isolate)


Download no_isolation.mw

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