SM 212 Final Examination
Fall 1991
 Find the explicit general solution to
 (a)

,
 (b)

,
 (c)

.
 (a)
 Use graphical analysis to sketch solutions to the differential
equation
corresponding to the four initial conditions
. Show correct slope and
concavity for each solution that you sketch. Also state
what the equilibrium solutions are.
Classify each equilibrium solution as stable or unstable.
 (b)
 Use two steps of Euler's method to find an approximation
to for the initial value problem
, .
 (a)
 Find the general solution to
 (i)

,
 (ii)

,
 (iii)

.
 (b)
 A damped massspring system hanging from a ceiling is
governed by the differential equation
.
 (i)
 From what point is the distance measured?
 (ii)
 State conditions on the parameters , , and which
will make the motion of the mass critically damped.
 (iii)
 The figure below shows an example of the motion of a massspring
system under each of the three types of damping. Which is which?
 Use the method of undetermined coefficients to solve the
initial value problem
 (a)
 Find
 (i)

,
 (ii)

,
 (iii)

, where
 (b)
 Use Laplace transoforms to solve
 (a)
 Use the convolution theorem to find
.
 (b)
 Find
.
Write your answer in summation notation.
What is ?
 (a)
 (i)
 A massspring system subject to an external force
is governed by the differential equation
Find the steady state motion of the spring.
 (ii)
 Write
in the form
.
What is the first positive value of for which is zero?
 (b)
 In the circuit shown below , , and stand for
the numerical values of the resistance, capacitance, inductance and electromotive
force, respectively. Find a system of differential equations
for the unknown current and the unknown charge on the capacitor.
Do not solve this system.
 Solve the initial value problem
 (a)
 For the differential equation
 (i)
 Find the recursive relation for the coefficients of a power
series solution
 (ii)
 Use the recursive relation in (a.i) to find the first three nonzero terms
of each of two linearly independent solutions to the differential equation.
 (b)
 What is the smallest radius of convergence that a power series solution
to the differential equation
could have?
 (c)
 (i)
 Find the Fourier series of the function
 (ii)
 State the values to which the Fourier series in (c.i) converges
at each of the points
.
 The temperature distribution on a thin bar satisfies
the following conditions
 (a)
 Interpret the boundary conditions physically.
 (b)
 Find . Show all steps of the separation of variables process
clearly. Write your answer in summation notation.
 (c)
 Use the first three terms of your answer to (b) to find the
termperature at the middle of the bar after two seconds.
David Joyner
20030801