1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

14 Chapter 0 Ordinary Differential Equations

For high-speed operation, the system is underdamped. Solve the initial

value problem consisting of the differential equation and the initial con-

ditionsy( 0 )=− 0. 001 h,y′( 0 )=0.

24.(Continuation) For an input speed of 25.4 m/s, it is observed thatσ∼=

600 Hz or 1200πradians/s andα= 0. 103 σ. Using these values, obtain

a graph of the solution of the preceding exercise, over the range 0<t<

0 .02 s. How far does the sheet move in 0.02 s?

25.(Continuation) The damping constantαreferred to in the previous ex-

ercises appears to depend onv, the speed of the sheet into the rollers,

according to the relationα/σ=A/v,whereAis a constant. From the in-

formation given previously, the value ofAis about 2.62. Assuming this is

correct, find the speedvat which damping is critical.

0.2 Nonhomogeneous Linear Equations

In this section, we will review methods for solving nonhomogeneous linear
equations of first and second orders,

du

dt

=k(t)u+f(t),

d^2 u

dt^2 +k(t)

du
dt+p(t)u=f(t).
Of course, we assume that the inhomogeneityf(t)is not identically 0. The
simplest nonhomogeneous equation is

du

dt=f(t). (1)

This can be solved in complete generality by one integration:

u(t)=

∫

f(t)dt+c. (2)

We have used an indefinite integral and have writtencas a reminder that there
is an arbitrary additive constant in the general solution of Eq. (1). A more
precise way to write the solution is

u(t)=

∫t

t 0

f(z)dz+c. (3)

Here we have replaced the indefinite integral by a definite integral with vari-
able upper limit. The lower limit of integration is usually an initial time. Note