#-------------------#
#   DdTF, Blatt 9   #
#-------------------#


# Aufgabe 1d)

xmax = 1.95
dx = 0.001

x = seq(0,xmax,by=dx)
nx = length(x)

y = rep(0,nx)

for(k in 2:nx)
{
  f = sqrt( 4/3*(y[k-1]+1)^2 - 1 )
  y[k] = y[k-1] + f*dx 
}

plot(x,y,asp=1,xlim=c(-3,3))    # asp = aspect ratio, x und y-Achse gleich skaliert



# Aufgabe 1e)

dx = 0.0001
x = seq(0,1,by=dx)
nx = length(x)

y = rep(0,nx)

for(y0prime in seq(0,1,by=0.01) )
{
  y = rep(0,nx)
 
  for(k in 2:nx)
  {
    f = sqrt( (1+y0prime^2)*(y[k-1]+1)^2 - 1 )
    y[k] = y[k-1] + f*dx 
  }
  print( c(y0prime,x[nx],y[nx]) ) 
}


y0prime = 1/3
y = rep(0,nx)
 
for(k in 2:nx)
{
  f = sqrt( (1+y0prime^2)*(y[k-1]+1)^2 - 1 )
  y[k] = y[k-1] + f*dx 
}

plot(x,y)