Initial conditions
(*c1 y[a] + c2 y’[a] = 0 = d1 y[b] + d2 y’[b] *)
In[]:=
ClearAll[c1,c2,d1,d2,a,b,x,y,μ,n]​​c1=1;c2=0;d1=1;d2=0;​​a=0;b=1;​​​​eqn=y''[x]+μy[x]==0;​​cond={c1y[a]+c2y'[a]==0,d1y[b]+d2y'[b]==0};​​sol=DSolve[{eqn,cond},y[x],x]/.

1
->1​​​​func[x_,n._,μ_]=y[x]/.sol;Manipulate[Plot[func[x,n,n^2Pi^2],{x,a,b}],{{n,1},0,20},SaveDefinitions->True]
Out[]=
y[x]
Sinx
μ

n.∈&&n.≥1&&μ
2
n.
2
π
0
True

Out[]=
​
n
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
In[]:=
y[x]/.sol
Out[]=

Sinx
μ

n.∈&&n.≥1&&μ
2
n.
2
π
0
True
