Awkward diff. eq., advice appreciated

  • Thread starter gstqtfr
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In summary, the conversation involves a person seeking help with solving an awkward differential equation. The equation is in the form of y' = x - xy -y and the person is struggling to find an integrating factor to solve it. There is confusion about whether the equation is to be solved with respect to x or t, and it is eventually determined that it is a system of two unknown functions and cannot be solved through a simple trick. The person is relieved to know this and plans to use numerical integration to find solutions.
  • #1
gstqtfr
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Hi,

I've got an awkward diff. eq. i'd really appreciate any help i can get with.

the DE is

y' = x - xy -y

in latex, it's

\frac{dy}{dt} & = & Cx - Dxy - Ey

but ignore the constants ...

it looked as though this would be easy to do with separation of variables at first, but couldn't work it out that way ... couldn't think of an integrationg factor ... it looks easy, but i seem to have a block on this, can't see a way ot get it into some useful form or find somehting to substitute that would make it come out ok ...

any ideas?
 
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  • #2
Do you know how to solve an equations such as

[tex] \frac{dy}{dx}+f(x) y= g(x) [/tex]

? In other words, do you know what an integrating factor is ?

Daniel.
 
  • #3
Hi Daniel,

Yes, I know what an IF is, but can't see an obvious one for this (i get the distinct feeling that there is one, but I'm damned if i can see it at the moment!)

one thing i didn't make at all clear when i posted the prob. is that we have

y'(t) = x - xy -y

apologies, it needs to be solved w.r.t. t

any suggestions on a suitable IF?
 
  • #4
help me out here, guys. am i missing anything mind-bogglingly obvious, such as an easy integrating factor, or is this actually a problem that requires something else? if anybody who vies this problem also can't see an obvious solution, let me know, will you? otherwise i'll think I'm just going nuts/suffering from advanced decripitude ...
 
  • #5
hello gstqtfr

By y'(t) do you mean [tex]\frac{dy}{dt}[/tex] ? Is x also a function of t and given in the question ? The question really isn't very clear. Can you post the question in its entire form ?
 
  • #6
Hi Arunbg,

Apologies, isn't clear at all, is it? BTW, 1st-time on this forum, can you post latex on here? anyway, the problem in 'tex is

\frac{dy}{dt} = Cx - Dxy - Ey

& both x and y are functions of t. this is why I'm finding this difficult to solve, i guess; if it were a case of

\frac{dy}{dx} = x - xy -y

(constants elided for clarity), then it's easy enough to separate the variables, for example. However, I can't see a way of doing this, since we have x(t) & y(t).

Any ideas? I'm really pretty stuck here ...
 
  • #7
In the way you imply it, it's ODE with 2 unknown functions. So it can't be really solved. I mean, determine BOTH x=x(t) and y=y(t) from the ODE.

Daniel.
 
  • #8
Hi Daniel,

Thanks for the reply, sorry I took so long to respond, I've been afk quite a bit today ...

Good! So it can't be solved through some trick I didn't know about. That's a relief - I was wondering whether there was some special technique I didn't know about (maybe related to PDEs, about which I know very little) that could be used.

Now I can get on to do some numerical integration, looking for steady states, etc.

Thanks for your help, guys!
 

FAQ: Awkward diff. eq., advice appreciated

What is an "awkward" differential equation?

An "awkward" differential equation typically refers to a differential equation that is difficult to solve using traditional methods. This could be due to complex or unusual functions, boundary conditions, or other factors.

How can I approach solving an awkward differential equation?

One approach is to try to simplify the equation by using substitutions or transformations. Another option is to use numerical methods, such as Euler's method or Runge-Kutta methods, to approximate a solution. It is also helpful to consult with a mathematics expert or use computer software to assist with solving the equation.

Are there any tips or tricks for solving awkward differential equations?

Some tips for solving awkward differential equations include breaking the problem down into smaller, more manageable parts, using symmetry to simplify the equation, and looking for known solutions that can be used as a starting point. It is also important to carefully consider the given boundary conditions and use them to guide your approach.

Are there any common mistakes to avoid when solving awkward differential equations?

One common mistake is to assume that there is only one "correct" way to solve an equation, which can limit your options and make the problem seem more difficult. It is important to be open to different approaches and to double-check your work for errors. Another mistake is to overlook important information, such as initial conditions or simplification techniques, that can help with solving the equation.

How can I check if my solution to an awkward differential equation is correct?

One way to check your solution is to plug it back into the original equation and see if it satisfies the equation. You can also use software or graphing tools to visualize your solution and see if it makes sense in the context of the problem. Another option is to compare your solution to known solutions or to consult with a mathematics expert for verification.

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