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