Does this differential equation have a closed form?

[euroazn]

I was busy doodling and basically ended up constructing this differential equation:

p'(t)=c(t)p(t)-c(t-T)p(t-T)

Obviously I've dealt with eq's like p'(t)=c(t)p(t) but I'm getting stuck because of the second term. Does this differential equation even have a closed form? Thanks.

 

[mfb]

For some c(t), it has solutions in a closed form. A general solution would be interesting, but I don't see one.

 

[Mute]

This is what's called a "delay differential equation". They are often much more difficult than regular differential equations, but depending on your choices for c(t) or other equations you want to investigate there may be some methods to deal with them analytically.

For example, if c(t) = const, you can try a solution of the form p(t) = exp(st). Plugging in this guess will give you a transcendental equation for s in terms of the Lambert-W function, giving you infinitely many possible solutions. I think that forming linear combinations of these solutions may enable you to fit any desired boundary conditions, but that's just a guess.

 

[euroazn]

This is what's called a "delay differential equation". They are often much more difficult than regular differential equations, but depending on your choices for c(t) or other equations you want to investigate there may be some methods to deal with them analytically.

For example, if c(t) = const, you can try a solution of the form p(t) = exp(st). Plugging in this guess will give you a transcendental equation for s in terms of the Lambert-W function, giving you infinitely many possible solutions. I think that forming linear combinations of these solutions may enable you to fit any desired boundary conditions, but that's just a guess.

Thank you! Now that I at least know the name of this type of equation I can probably figure out the solutions given a restricted set of c(t) myself.

EDIT: Or maybe not... it seems that constants for c(t) are about as good as it gets. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=878632&userType=inst

 

[blue_raver22]

I cannot provide a definitive answer without further context or information about the variables and constants involved. However, in general, a differential equation has a closed form if it can be expressed in terms of elementary functions such as polynomials, exponential functions, and trigonometric functions.

In the case of the given differential equation, it is possible that a closed form solution exists, but it may be difficult to find or express in terms of elementary functions. The presence of the second term, which includes a time delay, adds complexity to the equation and may require more advanced techniques for solving it.

I would recommend consulting with a mathematician or using numerical methods to approximate a solution, as finding a closed form solution may be challenging. Additionally, considering the specific application or context of the differential equation may also provide insights into the nature of the solution.