Where k and m are positive numbers what condition on k and

Question

(a) Consider the second order differential equation

(d²u/dt²)+k(du/dt)+mu=0

where k and m are positive numbers. What condition on k and m leads to critical damping?

Critical damping occurs when the roots of the characteristic equation

r²+kr+m=0

are equal; that is, when k²-4m=0.

(b) For the differential equation(d²u/dt²)+4u=17 sin(9t/4); u(0)=0, u'(0)=-4

show that beats occur.

The general solution is

u = c₁sin(2t) +c₂cos(2t) -16sin(9t/4)

and the initial conditions prescribe 0=c₂ and -4=2c₁-36 respectively, so that

u = 20sin(2t) -16sin(9t/4) = 4sin(2t)-16[sin(9t/4)-sin(2t)],

or u = 4sin(2t)-32sin(t/4)cos(17t/4)

(since by trig. sina-sinb=2sin(a-b)cos(a+b)) The latter term provides beats at frequency (1/4)l(2π) = 1/(8π).

(c) In the solution of u"-)4te+5u = 20sin(t) with u(0),), u'(0)=0,

what are the steady state and transient terms?

The solution (homogeneous plus particular terms) is u =-2e^-2tsin(t) + 2sin(t)

The particular term 2sin(t) is steady state and the homogeneous term -2e^-2tsin(t) is the transient term (decays to zero as t→∞).