Assigned August 2, 2022
Due August 9, 2022
MAT 136 H1S ASSIGNMENT 4
This assignment consists of two parts: a written component given below, and a WeBWorK component. It
is recommended that attempt at least some of the WeBWorK problems before the written problems. Your
answers to the written component must be structured in full sentences, and you must explain
your reasoning for each step of your computations.
Questions 1–3 concern a system of di?erential equations arising in chemical kinetics.
At time t (in seconds), a reaction mixture contains chemical species A and B in concentrations [A](t) and
[B](t) respectively (in mol/L). It is hypothesised that the concentrations satisfy the di?erential equations
d[A]
dt
= 1 3[A] + [A]2[B]
d[B]
dt
= 2[A] [A]2[B].
At time 0 seconds, the concentration of A is 3 mol/L and the concentration of B is 2 mol/L.
1. (a) Sketch the phase plane, drawing vectors at points ([A], [B]) for [A] 2 {0.5, 1, 1.5, 2} and [B] 2
{1, 1.5, 2, 2.5}. (You should draw 16 vectors in total.)
(b) Compute the equilibrium points.
2. Adapt the idea behind Euler’s method to approximate the concentrations of A and B at t = 0.3, using
3 steps.
3. Estimate the concentrations of A and B at t = 0.3 using the degree 2 Taylor polynomial approximations
of [A](t) and [B](t) around t = 0.
(Hint: compute [A]00 and [B]00. Actually only one of them requires work...)