(*Example on pattern formation discussed in class

Instability is achieved if the pair (λ,d) is such that 
there is some positive integer k such that d is larger than 
d_k(λ)

Caution: The domain of d_k(λ) consists of λ>k^2*Pi^2,     and thus the "left" branches of the graphs below should be excluded. 

Notice that as k increases, the graph quickly moves in the NE direction.
*)

Manipulate[Plot[2x^2+4k^2Pi^2/(k^2*Pi^2*(x-k^2*Pi^2)),{x,k^2*Pi^2-1,k^2Pi^2+1},AxesLabel->{λ,Subscript[d,k]}],{k,1,5}]

(*We se that if we pick (λ,d)=(10,300),then diffusive instability is achieved, but there is only 1 unstable mode since 
λ=10 only belongs to the domain of d_1 and not to any of the domains of d_k when k is larger than 1*)