I cannot solve the problem of the ant on a rubber rope (Ant on a rubber rope - Wikipedia, the free encyclopedia)

in the case of a rope stretching exponentially:

the target end of the rope y(t) goes away exponentially

y(t)= D*exp(H*t)

where D and H are positive constants. D is the initial length of the rope to be crossed by the ant.

Given that the rope stretches uniformly, I believe that the position of the ant on the rope x(t) should evolve according to

dx(t)/dt=c+x(t)*exp(H*t)

where c is the constant proper speed of the ant irrespective of rubber band stretching

initial condition x(0)=0

Can the ant reach the end of the rope depending on the relative values of c, D and H?

is it possible to prove that the ant will never reach the target, whatever the values of the constants?

or is it a critical point on the rope beyond which the target cannot be reached

many thanks in advance

Denis