Solution of varying coefficients ODE - r

Question

I have a set of observed raw data and use 2nd order ODE to fit the data
y''+b1(t)y'+b0(t)y = 0
The b1 and b0 are time-dependent and I use principal differential analysis(PDA) (R-package: fda, function: pda.fd)to get the estimate of b1(t) and b0(t) .
To check the validity of the estimates of b1(t) and b0(t), I use collocation method (R-package bvpSolve, function:bvpcol) to get the numerical solution of the ODE and compare the solution with the smoothing curve fitting of the raw data.
My question is that my numerical solution from bvpcol can caputure the shape of the fitting curve but not for the value of the function. There are different in term of some constant multiples.
(Since I am not allowed to post images,please see the link for figure)
See the figure of my output. The gray dot is my raw data, the red line is Fourier expansion of the raw data, the green line is numerical solution of bvpcol function and the blue line the green-line/1.62. We can see the green line can capture the shape but with values that are constant times of fourier expansion.
I fit several other data and have similar situation but different constant. I am wondering it is the problem of numerical solution of ODE or some other reasons and how to solve this problem to get a good accordance between numerical solution(green) and true Fourier expansion?
Any help and idea is appreciated!
Here is a raw data and code:
RData is here
library(fda)
library(bvpSolve)
# load the data
load('y.RData')
tvec = 1:length(y)
tvec = (tvec-min(tvec))/(max(tvec)-min(tvec))
# create basis
fbasis = create.fourier.basis(c(0,1),nbasis=nbasis)
bbasis = create.bspline.basis(c(0,1),norder=8,nbasis=47)
bfdPar = fdPar(bbasis)
yfd = smooth.basis(tvec,y,fbasis)$fd
yfdlist = list(yfd)
bwtlist = rep(list(bfdPar),2)
# PDA fit
bwt = pda.fd(yfdlist,bwtlist)$bwtlist
# output of estimated coefficients
beta0.fd<-bwt[[1]]$fd
beta1.fd<-bwt[[2]]$fd
# define the vary-coef function in terms of t
fbeta0<-function(t)eval.fd(t,beta0.fd)
fbeta1<-function(t)eval.fd(t,beta1.fd)
# define 2nd order ODE
fun2 <- function(t,y,pars) {
with(as.list(c(y,pars)),{
beta0 = pars[[1]];
beta1 = pars[[2]];
dy1 = y[2]
dy2 = -beta1(t)*y[2]-beta0(t)*y[1]
return(list(c(dy1,dy2)))
})
}
# BVP
yinit<-c(p1[1],NA)
yend<-c(p1[length(p1)],NA)
t<-seq(tvec[1],tvec[length(tvec)],0.005)
col<-bvpcol(yini=yinit,yend=yend,x=t,func=fun2,parms=c(fbeta0,fbeta1),atol=1e-5,islin=T)
# plot output
plot(col[,1],col[,2],col='green',type='l')
points(tvec,p1,col='darkgray')
lines(yfd,col='red',lwd=2)
lines(col[,1],col[,2],col='green',type='l')
lines(col[,1],col[,2]/1.62,col='blue',type='l',lwd=2,lty=4)
legend('topleft',col=c('green','darkgray','red','blue'),
legend=c('ODE solution','raw data','basis curve fitting','ODE solution/1.62'),lty=1)