Solve an eigenvalue problem with Dirichlet conditions:
In[]:=
sol=DSolve[{X''[x]+λ^2*X[x]==0,X[0]==0,X[1]==0},X[x],x]
Out[]=
X[x]

1
Sinx
2
λ

n.∈&&n.≥1&&
2
λ

2
n.
2
π
0
True

Make a table of the first 3 eigenfunctions:
In[]:=
eigfuns=TableX[x]/.sol[[1]]//.n.i,
2
λ
->
2
n.
2
π
/.{

1
1},{i,3}
Out[]=
{Sin[πx],Sin[2πx],Sin[3πx]}
Plot the eigenfunctions:
In[]:=
Plot[Evaluate[eigfuns],{x,0,Pi}]
Out[]=