Question 1 (10 points) Consider the differential equation dy
= t2 − y.
(a) Show that y(t) = ce−t + t2 − 2t + 2 is a solution to this equation for any value of c.
(b) Show that d
= 2t− t2 + y (Hint: differentiate the DE!).
(c) Consider a solution of the equation that passes through the point (2, 2). Use fact that
= t2 − y and the result in (b) to answer: At this point, is the solution increasing or
decreasing? Is it concave up or concave down? (You may want to verify your answer
on the Geogebra applet!)
Question 2 (8 points) Find a value of n for which the function y = x−1
is a solution of the
differential equation 1
. (Hint: plug in to both sides of the equation!)
Question 3 (8 points)
(a) Compute the integral
dx to show that it converges. Do not forget to write in
2 + x3/2
dx converge or diverge? Explain. (Hint: use part (a), and recall
that 0 ≤ sin2(x) ≤ 1.)
Question 4 (10 points) Recall that Hooke’s law says that force exerted by a spring extended
to length x is F (x) = kx, where k is the spring constant. Consider a spring that is extended
to 10cm by a mass of 2kg. How much work is done in extending the spring from this length
to length 25cm? You may assume that acceleration due to gravity is 9.8 m/s2. Be sure to
specify all units!
Question 5 (20 points) Consider the following differential equations:
= arccos(y)x2 (B) dy
= y + e−x (C) dy
= xe−y (E) dy
= x2 + y2
(a) Of these differential equations, exactly two are seperable. Circle those two above.
(b) Of the two seperable equations, one is very hard to solve, and the other much more
straightforward. Solve the straightforward one with initial value y(1) = 0.
(c) Of all the equations, only one has an equilibrium solution. Which is it? What is the
equilibrium solution? Is the equilibrium stable or unstable? (If you use a Geogebra
applet to help with the last part of this question, please paste an image from it here,
or at least sketch a sufficient amount of the slopefield to justify your answer.)
(d) One of the two slopefields below is for Equation B, and one is for Equation F. Label
which is which, and write below one distinguishing feature that enabled you to tell the
−2 −1 1 2
−2 −1 1 2
(e) Use Euler’s Method with ∆t = 2 to estimate y(5) for equation (F) given that y(1) = 0.
Is your answer an over or underestimate of the true value of y(5)? Explain briefly.
Question 6 (12 points) The density of pollution over a city at height h meters is well mod-
eled by p(h) = 0.0004e−0.0025h kg/m3 at altitude h meters. The city can be seen as a circle
of radius 8km, and the pollution measured in a cylinder with the city as its base.
(Note: this question is substantively the same as Q15 of the Varying Density lab. Feel
free to use your work from there here.)
(a) Using the function p(h) write a Riemann sum which approximates the amount of
pollutant in the cylindrical column of air up to height 1,000 meters, assuming we have
divided the cylinder into n slices of equal thickness. Label the volume of each segment,
and the density of pollutant on it.
(b) Write down an integral which yields the total amount of pollutant in the air over the
city to an altitude of 1,000 meters. (No need to compute the integral, just write it
(c) Pollution exists above 1,000 meters, of course. We could calculate it up to 2,000 meters,
or 3,000, but each of these would be arbitrary cutoffs. Write down and compute an
improper integral giving the total mass of pollutants over the city. (Hint: this is similar
to part (b). What should the new upper bound be?)
Question 7 (15 points) Consider the differential equation dy
= y2 − y − 2.
(a) By factoring the right hand side of the equation, find the equilibrium solutions of this
(b) Fill in the blank and cross out the incorrect answer: dy
is a degree polynomial
in y. Its top coefficient is positive/negative.
(c) Draw a graph of dy
vs. y below, labeling your axes and roots.
(d) Fill in the blanks (note that the first two should be the same): The graph in part (c)
has a critical point at y = . This implies that any solution that passes through
y = has an there.
(e) Use your graph to determine the direction of the ticks on a slopefield representing this
differential equation away from its equilibria. Make a sketch of the slopefield below,
indicating the equilibria, the direction of the ticks elsewhere, and the stability of each
equilibrium. Label your axes. Finally, sketch a solution that has the feature mentioned
in (d), and label that feature.
Question 8 (8 points) Let
3√x −1 ≤ x < 0
0 ≤ x < 1
f(x) dx. (Hint: you will need to split this up and compute two improper
integrals. Be sure to write and compute limits correctly!)
Question 9 (9 points) The following statement are all false. For each, explain why using
either calculation, worded explanation, or a counterexample, as appropriate.
(a) Any two solutions to a differential equation always differ by a constant (i.e. their
graphs are vertical shifts of each other).
(b) If F (x) is the force applied to an object (in Newtons) at distance x meters from its
origin, then the work done in moving the object a distance of d meters is W = Fd
(c) The function y(x) = Cex is the general solution of d