Is there a solution for these three differential equations?

In summary, the three differential equations have three unknowns, p, q, and r. If I replace f1 and f2 with 1 and 25.5, I can get a solution. However, Maple fails to find a solution for p, q, and r.
  • #1
Ad VanderVen
169
13
TL;DR Summary
I have given three differential equations with three unknowns and I cannot find a numerical solution with the 'fsolve' command in Maple. It seems like there is no solution at all. The question is whether this is the case, and if so, why.
I have three differential equations with three unknowns ##p##, ##q## and ##r##:

$$\displaystyle {\frac {\partial }{\partial p}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k \right) \right) =0$$,
$$\displaystyle {\frac {\partial }{\partial q}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k \right) \right) =0$$,
$$\displaystyle {\frac {\partial }{\partial r}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k \right) \right) =0$$
with
$$\displaystyle P \left( X=k \right) \, = \,{q}^{k-1}r+2\, \left( k-1 \right) {q}^{k-2} \left( 1-p-q-r \right) p+ \left( k-1 \right) {q}^{k-2}{p}^{2}+ \left( k-1 \right) {q}^{k-2} \left( 1-p-q-r \right) r\\
\mbox{}+ \left( k-1 \right) {q}^{k-2}pr+1/2\, \left( k-2 \right) \left( k-1 \right) {q}^{k-3} \left( 1-p-q-r \right) ^{3}+1/2\, \left( k-2 \right) \left( k-1 \right) {q}^{k-3} \left( 1-p-q-r \right) ^{2}p\\
\mbox{}+1/2\, \left( k-2 \right) \left( k-1 \right) {q}^{k-3} \left( 1-p-q-r \right) ^{2}r$$
and
##\displaystyle f_{{1}}\, = \,0##, ##\displaystyle f_{{2}}\, = \,26##, ##\displaystyle f_{{3}}\, = \,111##, ##\displaystyle f_{{4}}\, = \,17## and ##\displaystyle f_{{5}}\, = \,2##.

I can't find a solution for ##p##, ##q## and ##r##. Is there a solution at all?

If I replace ##\displaystyle f_{{1}}\, = \,0##, ##\displaystyle f_{{2}}\, = \,26## with ##\displaystyle f_{{1}}\, = \,1##, ##\displaystyle f_{{2}}\, = \,25## then I get a solution:

##p = 0.08557##, ##q = 0.05161##, ##r = 0.00641##.
 
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  • #2
These are not differential equations because in DEs the unknowns are functions (of time or some other independent variable). Here your unknowns seem to be numbers and the equations are algebraic. We just have to calculate some derivatives to find the final algebraic expression of the algebraic equation. So you should try a command in Maple that numerically solves algebraic equations.
 
  • #3
Be that as it may, there are three equations here with three unknowns ##p##, ##q## and ##r##. And of course I first used the 'solve' command to solve algebraically the equations in p, q and r. But that didn't work in Maple. So I tried to find a numeric solution with the command 'fsolve'. But that didn't work either and the question remains: why. Besides, you did not answer the question why I do get a solution when I change ##\displaystyle f_{{1}}\, =\,0## and ##\displaystyle f_{{2}}\, =\,26## to ##\displaystyle f_{{1}}\, =\,1## and ##\displaystyle f_{{2}}\, =\,25##.
 
  • #4
Well the full algebraic expression of the equations (after calculating the derivatives) is polynomial , however we seem to have 3 very complex polynomials with variables p,q,r and its no surprise to me that maple can't find a solution with algebraic expression (i mean a solution containing radicals , powers e.t.c). I think to this relates the fact that ##k## , which seems to be the variable controlling the degree of the polynomials, goes high up to 5, maybe if u set ##k## to be maximum 3 you ll get solutions with the solve command.

So numerical solution seems to me is the only working approach. When you change ##f_1## and ##f_2## you get a different system of polynomials for which there might be solutions. I am afraid i can't answer the question as to why when you do the specific change the system has solutions.
 
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  • #5
Thank you for your honest answer and for taking the time to answer my questions.
 
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  • #6
Two possible issues: (1) the solution is complex but the program is looking for solutions in the real numbers, (2) the numerical algorithm looking for solutions is starting its search at a particular point and doesn't find distant solutions. Can you control in Maple where the program starts looking for solutions? The one solution you did get does seem to be near zero, which is where I might expect the program to start searching by default.
 
  • #7
When I replaced ##\displaystyle f_{{1}}\, = \,0## and ##\displaystyle f_{{2}}\, = \,26## with ##\displaystyle f_{{1}}\, = \,1## and ##\displaystyle f_{{2}}\, = \,25## I got the following parameter estimates ##\hat{p} = 0.08557##, ##\hat{q} = 0.05161## and ##\hat{r} = 0.00641##. Using these estimates as starting values for the estimation based on the original ##f_k## values, Maple has also failed to find a solution.
 
  • #8
I would try things like ##f_1=0.5##, ##f_2=25.5## and see if you can narrow in on where you stop getting a solution, and what the solutions look like as you approach it.
 

Related to Is there a solution for these three differential equations?

1. What are differential equations?

Differential equations are mathematical equations that describe how a variable changes over time, based on the relationship between the variable and its rate of change. They are commonly used in physics, engineering, and other fields to model real-world systems and phenomena.

2. What types of solutions are there for differential equations?

There are two types of solutions for differential equations: explicit and implicit. Explicit solutions are expressed in terms of a specific variable, while implicit solutions are expressed in terms of the relationship between multiple variables.

3. Can differential equations always be solved?

No, not all differential equations have a solution that can be expressed in terms of known functions. In some cases, numerical methods must be used to approximate a solution.

4. How do you solve a system of differential equations?

There are several techniques for solving systems of differential equations, including separation of variables, substitution, and using matrix methods. The appropriate method depends on the specific equations and their complexity.

5. Can technology be used to solve differential equations?

Yes, technology such as computer software and calculators can be used to solve differential equations and obtain numerical solutions. However, it is still important to have a basic understanding of the underlying concepts and methods for solving differential equations.

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