An illustration of the Rayleigh-Ritz method to approximate eigenvalues.

f[x_,1]=x*(x-1)

f[x_,2]=x^2*(x-1)

f[x_,3]=x*(x-1)^2

f[x_,4]=x^2*(x-1)^2

f[x_,5]=x^3*(x-1)^2

f[x_,6]=x^2*(x-1)^3

f[x_,7]=x^3*(x-1)^3

Do[Do[a[i,j]=Integrate[D[f[x,i],x]*D[f[x,j],x],{x,0,1}],{i,1,7}],{j,1,7}]

Do[Do[b[i,j]=Integrate[f[x,i]*f[x,j],{x,0,1}],{i,1,7}],{j,1,7}]

A=Array[a,{7,7}]

MatrixForm[A]

B=Array[b,{7,7}]

MatrixForm[B]

Eigenvalues[{N[A],N[B]}]

N[Pi^2]

N[(2*Pi)^2]

N[(3*Pi)^2]

N[(4*Pi)^2]