#5) Taylor 5.22

Clear["Global`*"]

eq1= x''[t] + 2 β x'[t] + (w0^2) x[t] ==0

dsol= DSolve[eq1,x[t],t][[1]]

bc={x[0]==x0,x'[0]==v0}

eq2= Join[{eq1},bc]

dsol= DSolve[eq2,x[t],t][[1]] //FullSimplify

critical={β->w0};

xCrit[t_]=Limit[x[t] /.dsol,critical]

Plot[ xCrit[t] /.{x0->0, v0->1,w0->1},{t,0,5}]

Plot[ xCrit[t] /.{x0->0, v0->-1,w0->1},{t,0,5}]

Plot[{ xCrit[t],Exp[-t w0 ]} /.{x0->1, v0->0,w0->1} //Evaluate,{t,0,5},
PlotStyle->{Red,Blue,Green,Thick}]

xCrit[t]  /.{v0->0} //Simplify

xCrit[t]  /.{t->((2π)/w0),v0->0} //Simplify

xCrit[t]  /.{t->((2π)/w0),v0->0} //Simplify //N

xCrit[t]  /.{t->0,v0->0} //Simplify //N

#5) Taylor 5.22 The Wrong way!
Also See p.177: Critical Damping!!!

Clear["Global`*"]

eq1= x''[t] + 2 β x'[t] + (w0^2) x[t] ==0

dsol= DSolve[eq1,x[t],t][[1]]

dsol0=dsol /.{β->w0}

dsol0=dsol0 /.{C[2]->0}

eqX=x[0]==x[t] /.dsol0

eqX /.t->0

solX0=Solve[eqX /.t->0,C[1]][[1]]

eqV=v0==D[x[t] /.dsol0,t]

solV0= eqV /.t->0

solV0 /.solX0

Problem : if x[0] == 0, then v0 = 0.

#6) Taylor 5.41

Clear["Global`*"]

amp[w_] = ((f0^2)/(((((w0^2) - (w^2)))^2)+4 (β^2) (w^2)))

values={f0->1,w0->1,β->0.2};

Plot[amp[w] //.values //Abs  ,{w,0,2}]

eq1= (1/(2 ))amp[w0] == amp[w]

sol= Solve[eq1,w] //PowerExpand

Series[ w /.sol ,{β,0,1}]  //Normal//PowerExpand //Simplify

w/.sol  //PowerExpand //Simplify

((w/.sol  //PowerExpand //Simplify) /.{(w0^2)+(β^2)->(w0^2)} //PowerExpand //ExpandAll) /.{(β^2)->0}

(Series[#,{β,0,1}]& /@ %)  //PowerExpand //Normal//Simplify

Series[(Sqrt[1+ϵ]),{ϵ,0,2}]

#7) RLC

Clear["Global`*"]

xR=R;
xC=(1/(w c));
xL=w L;
z=Sqrt[(xR^2)+(((xL-xC))^2)]

current = (voltage/z)

sol1=Solve[D[current,w]==0,w]

maxCurrent= current /.{w->(1/((Sqrt[c]) (Sqrt[L])))} //PowerExpand

eq1 = current == (1/2) maxCurrent

sol2= Solve[eq1,w] //Simplify //PowerExpand

values1={R->10, c->20 (10^(-6)),L->200 (10^(-3)),voltage->200};
values2={R->20, c->20 (10^(-6)),L->200 (10^(-3)),voltage->200};

current /.{values1,values2} //Re

Plot[current /.{values1,values2} //Re //Evaluate,
{w,0,1000},
PlotStyle->{Red,Blue},
PlotRange->All,
GridLines->Automatic]

Plot[ {current /.values1,2 current /.values2} //Re //Evaluate,
{w,0,1000},
PlotStyle->{Red,Blue},
PlotRange->All,
GridLines->Automatic]

current /.values1

eq1

NSolve[eq1 /. values1,w]

545-458

NSolve[eq1 /. values2,w]

594-420

174/87

quality factor

((Sqrt[L/c])/R) /.values1

((Sqrt[L/c])/R) /.values2