Question

please solve 42.7: (a),(b),(c)

42.7. Consider = N()(a - bN(t – (m)] (a) Show that small displacements from equilibrium satisfy the following linear delay-differential equation: dN (1) = -aN (6- m. dt (b) Ifm > 0, we will not find the general solution. Instead, let us look for special solutions of the form of exponentials, N (1) = e. Show that r = -ae-rim

Answer 1

INI bata from the giren question @ using Taylor approximation, (45) (+) = f (Net), pit-tm)) + 511) topSC where f(w (t); plt-tm)) = N(tla-bp [t-tn)) » we get Slt) af 3t) + afs(t-tm) -la-bn it-tm)).8 (t) + -bp-sIt-tn) Now. N = 0 gives; S(t) = (a - b* ) Sit) -b- sit-tm) =-as (t-tm) or doi = -ap, Lt-tm) ( portaling s(t) = Ni(t) Putting 6 patting wilt) = ert, we have --ac rert a r(t-tm) rert = -aert=rtm ntm r = -ae

let for) = YA aerm Then for small tm, f(x)=0, has towo real roots; but for large tm, f(x)=0 has no real root. graphi tm=0.2 graph tm=0.3 zm=0.3 +