Avoiding loops in iterative solution of ODE - r

Question

I'm fitting a parametric model to some survival data with time-dependent covariates. The fitting procedure involves solving some ODEs iteratively - one ODE per time-interval per subject, but such that the initial condition for the ODE on the interval at hand is the last value of the solution to the ODE on the preceding interval. In that sense, the ODEs depend on each other.
My problem boils done to this: Right now, I'm solving these ODEs iteratively through a loop, since I need to use the last value of the previous solution as the starting point for the next. The problem is that this looping consumes a lot of time for large datasets. Is there some way in which I can use, say, vapply, or another vectorized function, to do the same thing?
I've been searching the archives, but nothing comes up as a solution to the problem of vectorizing an operation that depends on the previous value.
Here's a code example, that doesn't produce anything statistically meaningful on its own, but illustrates my problem:
require(odeSolve)
param <- c(a=1)
df <- function(t, state, param){
with( as.list(c(state, param)), {dX<-a*X; list(c(dX))} )
}
Data.i <- data.frame( lt=seq(0, 5, length=10)[-10],rt=seq(0, 5, length=10)[2:10], X=rnorm(9) )
Result <- vector(length=10)
Result[1] <- Data.i$X[1]
init <- c(X=Data.i$X[1])
for (k in 1:9){
t.seq <- seq(Data.i$lt[k],Data.i$rt[k],length=10)
sol <- as.numeric(ode(y = init, times = t.seq, func = df, parms = param)[10,-1])
Result[k+1] <- log(sol+X[k+1])
init <- c(X=sol)
}