Problem 1. Write down the equations for each of the reactants and products for the following reactions.
(a) A + 3B + C
k
−→ 2D + 2E.
(b) A
k1 −→ B + C
k2 −→ D.
Problem 2. Consider the following reaction
A
k1 −→ B
k2 −→ C.
For the following parts, use the link: https://www.desmos.com/calculator/srrpeadlou.
(a) Compare and contrast the reactions that take place given the three different scenarios
for initial conditions. Explain why what the graph displays makes sense and include
your graphs.
• [A]0 = 1, [B]0 = 0, and [C]0 = 0.
• [A]0 = 0, [B]0 = 1, and [C]0 = 0.
• [A]0 = 0, [B]0 = 0, and [C]0 = 1.
(b) For the initial conditions [A]0 = 1, [B]0 = 0, and [C]0 = 0, explain what happens when
you let
• k1 = 0 and k2 = 1,
• k1 = 1 and k2 = 0.
Include plots for these cases as well.
Problem 3. Using the same reaction as before:
A
k1 −→ B
k2 −→ C.
(a) Write a system of differential equations to models the reaction (i.e. write an ODE to
model, [A], [B], and [C].
(b) Solve your three ODE’s using the initial conditions [A]0 = 1, [B]0 = 0, and [C]0 = 0 and
k1 = 1, k2 = 1.
Problem 4. If x1(t) and x2(t) are solutions to the differential equation
x
′′ + bx′ + cx = 0
is x = x1 + x2 + k for a constant k always a solution? Is the function y = tx1 a solution?
Show why or why not.
1
Problem 5. Consider the following initial value problem:
x
′′ + 4x
′ + 3x = 0
with initial data x(0) = 1, x
′
(0) = 0.
(a) Find the solution.
(b) Sketch a plot of the solution.
(c) Explain in words what is happening to the solution as t → ∞?
Problem 6. Consider the following differential equation:
x
′′ + 2x
′ + x = sin(t)
(a) Find the homogeneous solution xH(t).
(b) Find the particular solution xP (t).
(c) Find the specific solution corresponding to the initial data x(0) = 0, x
′
(0) = 0.
(d) Plot the curve (x(t), x′
(t)) in the plane (we often call this phase space). Use this link here:
https://www.desmos.com/calculator/ouqwcxj2xz. Use the time range t ∈ [0, 10π].
(e) Describe what happens with this system over time. Does it seem to approach some kind
of stable solution? Note that this stable solution could be periodic.
Problem 7. Optional for up to +5 extra credit: Write down a homogeneous secondorder linear differential equation where the system displays a decaying oscillation. For full
credit, justify why we should expect this system to behave the way it does.
