DSolve[{x'[t]==x[t],x[0]==1},x[t],t]

DSolve[{x'[t]==y[t],y'[t]==x[t]},{x[t],y[t]},t]

sol=DSolve[{x'[t]==y[t],y'[t]==x[t],x[0]==1,y[0]==-1},{x[t],y[t]},t]

Plot[{x[t]/.sol[[1]],y[t]/.sol[[1]]},{t,0,1}]

sol[[2]]

sol=NDSolve[{x'[t]==y[t],y'[t]==x[t],x[0]==1,y[0]==-1},{x[t],y[t]},{t,0,1}]

Plot[{x[t]/.sol[[1]],y[t]/.sol[[1]]},{t,0,1}]

(* 
u(x,0)=f[x];
u(0,t)=ϕ[t], u(1,t)=ψ(t) *)

ϕ[t_]:=0;
ψ[t_]:=0;
f[x_]:=x(x-1);
n=4
Δx=1.0/n
eqs=Table[D[Subscript[u,r][t],{t,1}]==Subscript[u,r+1][t]-2Subscript[u,r][t]+Subscript[u,r-1][t],{r,1,n-1}]/.{Subscript[u,0][t]->ϕ[t],Subscript[u,n][t]->ψ[t]};
coi=Table[Subscript[u,r][0]==f[r Δx],{r,1,n-1}];

eqs

coi

difeqs=Join[eqs,coi]