In[]:=
λ[n_]=1/4+n^2;​​
In[]:=
Y[x_,n_]=Exp[-1/2*x]*Sin[n*x]​​
Out[]=
-x/2

Sin[nx]
In[]:=
Integrate[Y[x,1]*Y[x,2],{x,0,Pi}]​​
Out[]=
1
5
(1+
-π

)
In[]:=
norm[n_]=Sqrt[Integrate[Exp[x]*Y[x,n]^2,{x,0,Pi}]]​​
Out[]=
π
2
-
Sin[2nπ]
4n
In[]:=
Simplify[%,n∈Integers&&n>0]
Out[]=
π
2
In[]:=
X[x_,n_]=Y[x,n]/norm[n];
In[]:=
Integrate[Exp[x]*X[x,1]*X[x,2],{x,0,Pi}]
Out[]=
0
In[]:=
f[x_]=x*(Pi-x)
Out[]=
(π-x)x
In[]:=
fig1=Plot[f[x],{x,0,1},PlotStyle->{Thick},PlotRange->All]
Out[]=
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0
In[]:=
A[n_]=Integrate[Exp[x]*f[x]*X[x,n],{x,0,Pi}]
Out[]=
(-32n(6+4
2
n
(-2+π)+π)+8
π/2

(-4n(-6+π+4
2
n
(2+π))Cos[nπ]+(-4+48
2
n
+π-16
4
n
π)Sin[nπ]))
3
(1+4
2
n
)
2π-
Sin[2nπ]
n
In[]:=
Simplify[%,n∈Integers&&n>0]
Out[]=
-
16n
2
π
6+4
2
n
(-2+π)+π+
n
(-1)
π/2

(-6+π+4
2
n
(2+π))
3
(1+4
2
n
)
In[]:=
S[x_,N_]:=Sum[A[n]*X[x,n],{n,1,N}];
In[]:=
fig2=Plot[S[x,5],{x,0,1},PlotStyle{Red,Dashed}]
Out[]=
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0
In[]:=
fig3=Plot[S[x,10],{x,0,1},PlotStyle{Green}]
Out[]=
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0
In[]:=
fig4=Plot[S[x,20],{x,0,1},PlotStyle{Black,Dashed}]
Out[]=
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0
In[]:=
Show[fig1,fig2,fig3,fig4]
Out[]=
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0