12 Chapter 0 Ordinary Differential Equations
EXERCISES
In Exercises 1–6, find the general solution of the differential equation. Be care-
ful to identify the dependent and independent variables.
- d
(^2) φ
dx^2
+λ^2 φ=0.
2.
d^2 φ
dx^2 −μ(^2) φ=0.
- d
(^2) u
dt^2 =0.
- dT
dt
=−λ^2 kT.5.
1
rd
dr(
rdw
dr)
−
λ^2
r^2 w=0.- ρ^2 d
(^2) R
dρ^2
- 2 ρdR
dρ
−n(n+ 1 )R=0.
In Exercises 7–11, find the general solution. In some cases, it is helpful to carry
out the indicated differentiation, in others it is not.
- d
dx
(
(h+kx)dv
dx)
=0(h,kare constants).- (exφ′)′+λ^2 exφ=0.
- d
dx
(
x^3 du
dx)
=0.
10.r^2 d(^2) u
dr^2
+rdu
dr
+λ^2 u=0.
11.^1
r
d
dr(
rdu
dr)
=0.
12.Compare and contrast the form of the solutions of these three differential
equations and their behavior ast→∞.a. d(^2) u
dt^2 +u=0; b.
d^2 u
dt^2 =0; c.
d^2 u
dt^2 −u=^0.
In Exercises 13–15, use the “exponential guess” method to find the general
solution of the differential equations (λis constant).
13.
d^4 u
dx^4 −λ
(^4) u=0.
- d
(^4) u
dx^4 +λ
(^4) u=0.