Exercise 2.5.1: Find a particular solution of y + 5y + 6y = 2x+ 1.
Exercise 2.5.2: Find a particular solution of y' - y' - 6y = e
Exercise 2.5.3: Find a particular solution of yr - 4y' + 4y = 2
Exercise 2.5.4: Solve the initial value problem y + 9y = cos(3x) + sin(3x)for y(0) = 2, y'(O) = 1.
Exercise 2.5.5: Set up the form of the particular solution but do not solve for the coefficients for
y(4-2y" + y" = e'.
Exercise 2.5.6: Set up the form of the particular solution but do not solve for the coefficients for
y(4) - 2y" + y" = er sin x.
Exercise 2.5.7: a) Using variation of parameters find a particular solution of Y' 2y +y = e*.
b) Find a particular solution using undetermined coefficients. c) Are the two solutions you found
the same? See also Exercise 2.5.10.
Exercise 2.5.8: Find a particular solution of y" - 2y = sin(x). It is OK to leave the answer as
a definite integral.
Exercise 2.5.9: For an arbitrary constant C find a particular solution to y" - y = ecs. Hint: Make
sure to handle every possible real C.
Exercise 2.6.1: Derive a formula for xap if the equation is mx" cx' + kx = Fosin(wor). Assume
Exercise 2.6.2: Derive a formula for xap if the equation is mx" +cx' +kx = Fo cos(wr)+ F1 cos(3ax).
Assume C > 0.
Exercise 2.6.3: Take mx" + cx' + kx = Fo cos(wot). Fix III > 0, k O, and Fo > 0. Consider the
function C(w). For what values of c (solve in terms of m1, k, and Fo) will there be no practical
resonance (that is, for what values of c is there no maximum of C(w) for w > 0)?
Exercise 2.6.4. Take mx" + cx' + kx = Fo cos(wot). Fixc > O, k O. and Fo > 0. Consider the
function C(w). For what values of 111 (solve in terms of c, k. and Fo) will there be no practical
resonance (that is, for what values of m is there no maximum of C(w) for w > 0)?
Exercise 2.6.5: Suppose a water tower in an earthquake acts as a mass-spring system. Assume
that the container on top is full and the water does not move around. The container then acts as a
mass and the support acts as the spring, where the induced vibrations are horizontal. Suppose that
the container with water has a mass of III = 10,000kg. It takes a force of 1000 newtons to displace
the container I meter. For simplicity assume no friction. When the earthquake hits the water tower
is at rest (it is not moving).
Suppose that an earthquake induces an external force F(t) = mAw² cos(wot).
a) What is the natural frequency of the water tower?
b) If is not the natural frequency, find a formula for the maximal amplitude of the resulting
oscillations of the water container (the maximal deviation from the rest position). The motion will
be a high frequency wave modulated by a low frequency wave, so simply find the constant in front
of the sines.
c) Suppose A = 1 and an earthquake with frequency 0.5 cycles per second comes. What is the
amplitude of the oscillations? Suppose that if the water tower moves more than 1.5 meter from the
rest position, the tower collapses. Will the tower collapse?
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