Setup:

Needs["Calculus`VectorAnalysis`"]
Needs["Calculus`DSolve`"]
Needs["Algebra`Trigonometry`"]
Needs["Graphics`Graphics`"]
Needs["Graphics`Graphics3D`"]
Needs["Graphics`ParametricPlot3D`"]

Example 1a: Cartesian

Clear["Global`*"]

SetCoordinates[Cartesian]

psi[x,y,z] = X[x] Y[y] Z[z];

eq1= Laplacian[ psi[x,y,z] ]/psi[x,y,z] //Expand

select[eq_,var_]:= Select[eq,(!FreeQ[#,var])&]

xEq= select[eq1,x] == -cx^2

yEq= select[eq1,y] == -cy^2

zEq= select[eq1,z] == +cz^2

Note: there is a constraint:  -cx^2 -cy^2 +cz^2 ==0

xEq2= Thread[xEq X[x],Equal]

xSol= DSolve[xEq2,X[x],x][[1]]

xSol //ComplexToTrig //Simplify

zEq2= Thread[zEq Z[z],Equal]

zSol= DSolve[zEq2,Z[z],z][[1]]

Example 1b: Cylindrical

Clear["Global`*"]

SetCoordinates[Cylindrical]

psi[r,theta,z] =R[r] Theta[theta] Z[z];

Example 1c: Spherical

Clear["Global`*"]

SetCoordinates[Spherical]

psi[r,theta,phi] =R[r] ang[theta,phi];

We will show the solution of this equation is SphericalHarmonicY

rule= {ang -> (SphericalHarmonicY[el,m,#1, #2]&)}

eq4= eq3 /.rule //Simplify

We do this for a range of {el,m} and assume it generalizes.

Table[ eq4 ,{el,1,6},{m,0,el-1}] //Simplify //N //Chop

Example 4:  SphericalHarmonicY

Clear["Global`*"]

plot[l_,m_]:=
SphericalPlot3D[ 
       SphericalHarmonicY[l,m,theta,phi] //Abs //Evaluate
      ,{theta,0,Pi}
      ,{phi,0,2 Pi}
      ,Boxed->False
      ,Axes->False
      ];

plot[5,1];

Do[
Do[

Print["{l,m}=",{l,m}];
plot[l,m];

,{m,0,l}]
,{l,0,3}
]

Problem 5: SphericalHarmonicY
(I've done the example here with Sin. You need to change it.
This serves as a templave for many of the other problems.)

Clear["Global`*"];

Note: the conjugate of  SphericalHarmonicY[2,m, theta, phi]   
= (-1)^m SphericalHarmonicY[2,-m, theta, phi]

doit[m_,n_]:= 
Integrate[  Sin[m theta]  Sin[n theta]  ,{theta,0,2 Pi} ]

Table[ doit[n,m] ,{n,0,3},{m,0,3}] //TableForm

Plot[ Table[ Sin[n theta] ,{n,0,3,1}] //Evaluate 
      ,{theta,0, 2 Pi}
      ,PlotStyle->Table[Thickness[0.004 * i],{i,1,4}]
      ];

Problem 6: Rayleigh's Formula
Choose k=1, and pick a range for the integration that looks nice. 
Take about 15 terms in the seties

Clear["Global`*"];

Problem 7: LegendreP

Clear["Global`*"];

Problem 8: Expansion in LegendreP
z=Cos[theta]

Clear["Global`*"];

Problem 9: Expansion in LegendreP
z=Cos[theta]

Problem 10:

Clear["Global`*"];

(* This is the preferred weighting to make "nice" plots *)
Do[

Print["n=",n];
Plot[ (HermiteH[n,x] E^(-x^2/2))^2 ,{x,-6,6},PlotRange->All];
      
,{n,0,3}];

Problem 11:

Clear["Global`*"];

sphBessel[n_,z_]= Sqrt[Pi/(2 z)] BesselJ[n+1/2,z]