Chapter 0 Ordinary Differential Equations 13
- d
(^4) u
dx^4
- 2 λ^2 d
(^2) u
dx^2
+λ^4 u=0.
In Exercises 16–18, one solution of the differential equation is given. Find a
second independent solution.
- d
(^2) u
dt^2 +^2 a
du
dt+a
(^2) u=0, u 1 (t)=e−at.
17.t^2 d
(^2) u
dt^2
+( 1 − 2 b)tdu
dt
+b^2 u=0, u 1 (t)=tb.
- d
dx
(
xdu
dx)
+^4 x(^2) − 1
4 x
u=0, u 1 (x)=cos√(x)
x
.
In Exercises 19–21, use the indicated change of variable to solve the differential
equation.
- d
dρ
(
ρ^2 dR
dρ)
+λ^2 ρ^2 R=0, R(ρ)=u(ρ)
ρ.
20.
d
dρ(
ρdφ
dρ)
+
4 λ^2 ρ^2 − 1
4 ρ φ=0, φ(ρ)=v(ρ)
√ρ.21.t^2 d(^2) u
dt^2
+ktdu
dt
+pu=0, x=lnt, u(t)=v(x).
22.Solve each initial value problem. Assuming that the solution represents
the displacement of a mass in a mass–spring–damper system, as in the
text, describe the motion in words.
a. d
(^2) u
dt^2
- 4 u=0, u( 0 )=1, du
dt
( 0 )=0;
b.d^2 u
dt^2 +^2du
dt+^2 u=0, u(^0 )=1,du
dt(^0 )=1;c. d(^2) u
dt^2 +^2
du
dt+u=0, u(^0 )=1,
du
dt(^0 )=1;
d. d
(^2) u
dt^2
- 2 du
dt
- 75 u=0, u( 0 )=0, du
dt
- 75 u=0, u( 0 )=0, du
( 0 )=1.
23.Sheet metal is produced by repeatedly feeding the sheet between steel
rollers to reduce the thickness. In the article “On the characteristics and
mechanism of rolling instability and chatter” [Y.-J. Lin et al.,J. of Manu-
facturing Science and Engineering, 125 (2003): 778–786], the authors find
that the distance between rollers is well approximated byh+y,whereh
is the nominal output thickness andyis the solution of the differential
equationy′′+ 2 αy′+σ^2 y=0. The elasticity of the sheet and the rollers
provides the restoring force, and the plastic deformation of the sheet ef-
fectively provides damping.