Problem #1 :

Clear["Global`*" ]

m= {{7,3},{3,-1}};
m //MatrixForm

Eigenvalues[m]

ev= Normalize /@ Eigenvectors[m]

ev.m.Transpose[ev] //Simplify

Problem #2 :

Clear["Global`*" ]

kin=(1/2)(m1 (R^2)((θ1'[t])^2)+m2 (R^2)((θ2'[t])^2)+m3 (R^2)((θ3'[t])^2))

pot= (1/(2 ))k (R^2)((((θ1[t]-θ2[t]))^2)+(((θ2[t]-θ3[t]))^2)+(((θ3[t]-θ1[t]))^2))

pot //Expand //Collect[#,k (R^2)]&

lag=kin-pot

D[lag,θ1[t]] //Simplify

D[D[lag,θ1'[t]],t]

eq1= D[lag,θ1[t]]-D[D[lag,θ1'[t]],t] //Simplify

eq2=D[lag,θ2[t]]-D[D[lag,θ2'[t]],t] //Simplify

eq3=D[lag,θ3[t]]-D[D[lag,θ3'[t]],t] //Simplify

eqs= {eq1,eq2,eq3} /.{m1->m,m2->m,m3->m};
eqs //MatrixForm

-eqs /.{m->0,R->1,k->1} //MatrixForm

vmat={{2,-1,-1},{-1,2,-1},{-1,-1,2}};
k vmat //MatrixForm

-eqs /.{m->1,R->1,k->0} //MatrixForm

tmat= DiagonalMatrix[{1,1,1}];
m tmat //MatrixForm

mat= vmat - w2 tmat;
mat //MatrixForm

Det[mat]

Solve[Det[mat]==0,w2]

Problem #3

Clear["Global`*"]

term = (Sqrt[((((x'[y])^2)+1)/y) ])

D[term,x[y]]

D[D[term,x'[y]],y]

tmp2=D[term,x'[y]] //PowerExpand

eq=tmp2^2 == (1/(2 a))

sol= Solve[eq,x'[y]]

tmp4=x'[y] /.sol[[2]]

jacobian= D[a(1-Cos[θ]),θ]

tmp5=tmp4  jacobian /.{y->a(1-Cos[θ])}

tmp6=(Sqrt[tmp5^2]) //FullSimplify //PowerExpand

tmp7 =tmp6 //TrigExpand //FullSimplify

tmp8= Integrate[ tmp7,θ]

y[θ_]=-tmp7
x[θ_]=tmp8

ParametricPlot[{x[θ],y[θ]} /.{a->1},{θ,0,6π}]

Problem #4 :

Clear["Global`*" ]

kin= (1/2)m1 ((x'[t])^2)+(1/2)α m2 (R^2) ((θ'[t])^2)

pot=- m1 g x[t]

lag=kin-pot

constraint = x[t] - R θ[t]

{D[constraint,x[t]],D[constraint,θ[t]]}

eqx= D[lag,x[t]]-D[D[lag,x'[t]],t] +λ D[constraint,x[t]]==0 //Simplify

eqθ= D[lag,θ[t]]-D[D[lag,θ'[t]],t] +λ D[constraint,θ[t]]==0 //Simplify

eqF =D[constraint,{t,2}] ==0

sol= Solve[ {eqx,eqθ,eqF},{x''[t],θ''[t],λ}] //Simplify

sol /.{α->0}

Problem #5 :

Clear["Global`*" ]

Integrate[ DiracDelta[x-2],{x,0,Infinity}]

Integrate[ DiracDelta[2x-2],{x,0,Infinity}]

Integrate[(5x-2) DiracDelta[x-2],{x,0,Infinity}]

Integrate[ϕ[x] DiracDelta[((x^2)-(a^2))],{x,0,Infinity},Assumptions->{a>0}]

Integrate[ Cos[x] DiracDelta[Sin[x]],{x,((-π)/2),(π/2)}]

Integrate[ DiracDelta[Cos[x]],{x,0,2π}]

Integrate[ DiracDelta[Cos[x]],{x,0,4π}]

Integrate[ ϕ[x]DiracDelta'[x],{x,-Infinity,Infinity}]

Problem #6 :

Clear["Global`*" ]

Integrate[g[x] h[y] DiracDelta[x+y-z],{x,0,1},{y,0,1},Assumptions->{1>z>0,z>x}]//Expand

g[x_]=Exp[-x] - Exp[-2x];
h[y_]=Exp[-y];

Integrate[g[x] h[y] DiracDelta[x+y-z],{x,0,1},{y,0,1},Assumptions->{1>z>0}]//Expand

Integrate[g[x] h[z-x]  ,{x,0,z}, Assumptions->{1>z>0}]//Expand

Integrate[g[z-y] h[y]  ,{y,0,z}, Assumptions->{1>z>0}]//Expand

Problem #7

term[n_,x_,y_]:=  Sin[n π x] Exp[- n π y]

Plot3D[ term[1,x,y],{x,0,1},{y,0,1},PlotRange->All]

c[n_]=Integrate[  Sin[n π x]  ,{x,0,1}]

Limit[c[n],n->0]

c[0]=0

sum[m_,x_,y_]:=Sum[ c[n] term[n,x,y],{n,0,m}]

Plot3D[ sum[30,x,y],{x,0,1},{y,0,1},PlotRange->All]