Description
1) Solve the following homogeneous equations
a)
y
00 + 3y
0 + 2y = 0
b)
y
000 = o
c)
y
(4) + 8y
00 + 16y = 0
2) Solve the following IVP
y
00 + 2y
0 + 4y = 0 y(0) = 1, y0
(0) = −1 + 2√
3
3) When a spring is stretched or compressed, its restoring force is directly
proportional to its change in length. If x represents the displacement
of the weight from its equilibrium position, then by Hooke’s Law
F = −kx k > 0
where the minus sign indicates that the restoring force F is always opposite in direction to the displacement. Combining this with Newton’s
Second Law gives,
m
d
2x
dt2
= −kx
or
m
d
2x
dt2
+ kx = 0
a) If the spring is initially displaced to x(0) = 5 and it’s initial velocity is zero (i.e. x
0
(0) = 0), find x(t) for t > 0.
b) Sketch your solution in a).
c) This approximation is decent, but doesn’t really reflect what happens in real life; in reality, the amplitude of the oscillations decreases until eventually the mass is at rest. Thus, we need to
include another term which acts as air resistance. At slow speeds,
the force of air resistance on an object is well approximated by
Fa = −c
dx
dt , where c > 0 and dx
dt is the speed of the object. Thus
our differential equation becomes
m
d
2x
dt2
+ c
dx
dt + kx = 0
Assume c
2 < 4mk and find x(t) in terms of c, m and k assuming
the same initial conditions as in a).
d) Sketch your solution in c).
4) Find the general solution of
y
000 + 4y
0 = x
2 + sinx
2