Exercises
1. An over-damped spring/mass system is modeled by the DE
d2 x
dx
+ 5 + 4x = 0
2
dt
dt
with x(0) = x0 (0) = 1. The problem can be interpreted as representing the over-damped motion of a mass on
a spring. The mass starts from a position 1 unit below the equilibrium position with a downward velocity of 1
ft/s. Solve the DE and interpret the motion. Is the system in steady or transient state?
2. A driven spring/mass with damping is modeled by
dx
1 d2 x
+ 1.2 + 2x = 5cos4t.
5 dt2
dt
Given that the initial displacement is 0.5 and the initial velocity is 0. Solve the system and determine if the
system is in steady or transient state.
3. In an RLC circuit, the discharge over the capacitor can be modeled by the DE
R
dq 1
+ q = 2(1 − e−t ),
dt C
where R = 1Ω and C = 0.5F. Solve the IVP if the initial condition is q(0) = 1 and discuss the solution as
t → ∞.
4. A charge q in an RLC circuit can be modeled by the DE
L
d2 q
dq 1
+ R + q = 0,
2
dt C
dt
where L = 2, R = 3 and C = 0.1. Solve the IVP if the initial conditions are q(0) = 1 and q0 (0) = 0.
5. A vehicle rests on a spring-shock absorber system on each of its four wheels. Assume
25y00 (t) + 50y0 (t) + 425y = 25e−t cos4t
to be a model that describe the impact on the vehicle due to a bumpy road. Let y(0) = y0 (0) = 0 and solve the
IVP.
6. An undriven spring/mass system has no external force f (t). Find the solution that represents the harmonic
motion of the damped system
d2 x
dx
+ 4 + 8x = 0,
2
dt
dt
if the initial displacement is x(0) = and the initial velocity applied to the mass is x0 (0) = 0. Is the system in
steady or transient state.
7. A driven spring/mass system has an external force f (t) = 4sin2t. Find the solution that represents the harmonic
motion of the damped system
d2 x
dx
+ 4 + 4x = 5sin2t,
dt
dt2
subject to an initial displacement x(0) = 1 and an initial velocity of x0 (0) = 0. Is the system in steady or
transient state?
8. Mpho is smoothly driving his Polo Vivo along the N1 highway, on his way home to Limpopo. At t=2 hours, he
hits a pothole. That is, a sudden shock is applied to an arbitrary system (the system hypothetically models the
motion of his vehicle)
θ00 (t) + 2θ0 (t) + 2θ = 5 f (t)
with initial displacement θ(0) = 1 and initial velocity θ0 (0) = 0. Find an expression for the displacement θ in
terms of t. Discuss his motion after hitting the pothole.
c Compiled by SM Simelane
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