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Question: How to integrate this ode and how to substitue this boundary value

Question:How to integrate this ode and how to substitue this boundary value

C_R 2847 Maple
integration ode differential-equations bvp boundary-conditions
June 03 2023
1

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 {} ?

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