Van Meegern Art Forgeries

Question 1.
The equation is separable so we have
       dN/N = - lambda * dt
  which implies
      ln (N) = - lambda*t + C   or   N = C e^(- lambda*t)
Since N(t_0)=N_0 we have
       N(t_0) = C e^(-lambda*t_0) = N_0   =>   C = N_0 * e^(lambda*t_0)
Thus
      N(t) = N_0* e^(-lambda*t +lambda*t_0)

DSolve[ {y'[t]  ==  -lambda * y[t],  y[0] == y0}  ,
       y[t], t]

DSolve[ {y'[t]  ==  -lambda * y[t],  y[t0] == y0}  ,
       y[t], t]

Question 2.

      .5*N_0 = N_0* e^(-lambda*t +lambda*t_0) => .5 = e^(-lambda*t)
Solving for t, we get
       ln(.5) = -lambda * t  =>   t = -ln(.5)/lambda = -ln(1/2) / lambda = -(ln 1 - ln 2)/lambda
                                                 = ln 2 / lambda

Solve[ .5  ==  E^(-lambda*t),  t]

Question 3.
An integrating factor of e^(lambda*t) will make the equation integrable.
         (e^(lambda*t) y' = (-lambda*y + r) e^(lambda*t)
         d/dt ( e^(lambda*t)y ) = (e^(lambda*t))*y' + lambda*(e^(lambda*t))*y
Thus
          d/dt ( e^(lambda*t)y ) = r *( e^(lambda*t)  )
          e^(lambda*t)y = (r/lambda) * e^(lambda*t) + C
or
          y(t) =C e^(-lambda*t) + r / lambda
  If  y(t0)=y0 then
         y0 = C  e^(-lambda*t0) + r / lambda
  and thus
        C=(y0 -r / lambda) * e^(lambda*t0)

DSolve[ {y'[t]  ==  -lambda * y[t] + r }  , y[t], t]DSolve[
        {y'[t]  ==  -lambda * y[t] + r , y[t0] == y0}  , y[t],
       t]

Question 4.

We can write the solution as
      y(t) =[(y0 -r / lambda) * e^(lambda*t0)  ]* e^(-lambda*t) + r / lambda
            = e^(lambda*(t0-t)) * (y0-r/lambda) + r/ lambda
             =e^(-300 lambda ) * (y0-r/lambda) + r/ lambda
or
     lambda* y = e^(-300 lambda ) *(lambda*y0 - r)  + r

Solving for    lambda* y0 we have
         lambda * y0 =  e^(300 lambda ) * (lambda*y) - r * e^(-300 lambda ) + r

f[pb_, ra_] :=        pb*E^(300*lambda) - ra*(E^(300*lambda) - 1)\(		lambda = Log[2]/22; 
y1 = f[8.5, 0.8]\)y2 = f[12.6, .26]y3 = f[1.5, 1.4]y4 = f[10.3, .3]y5 = f[8.2, 5.5]y6 = f[5.5, 6.0]p1 = ListPlot[{y1, y2, y3, y4, y5, y6}, PlotJoined -> True]p2 =
       Plot[30000, {x, 1, 6}, PlotStyle -> RGBColor[1,  0.221012,  0.180026]]
         Show[p1, p2,
       Graphics[Text["Lace Maker", {3, 100000}, {-1, 0}, {0, -1}]
            ]]