Name:
• I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion
of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.
• Use correct mathematical notation.
1. Consider the partially decoupled system
dx
dt = x + 2y + 3z (1)
dy
dt = 4y + 5z (2)
dz
dt = 6z. (3)
(a) (1 point) Find the equilibrium point(s) of the system.
(b) (7 points) Derive the general solution.
(c) (2 points) Find the solution that satisfies the IC (x0, y0, z0) = (2, 5, 6).
2. Euler’s Method for Autonomous System1 As we saw in Section 1.4, Euler’s method for a first-order equation is based
on the idea of approximating the graph of a solution by line segments whose slopes are obtained from the differential
equation. Euler’s approximation scheme for systems of differential equations works essentially the same way! The
following is the summary of Euler’s method for autonomous systems:
Euler’s Method for Autonomous Systems
Given the autonomous systems
dx
dt = f(x, y) (4)
dy
dt = g(x, y), (5)
the initial condition
x(0), y(0)
= (x0, y0), and the step size ∆t, compute the point (tk+1, xk+1, yk+1) from the
preceding point (tk, xk, yk) as follows:
Step 1. Use the right-hand sides of (4) and (5) to compute both f(xk, yk) AND g(xk, yk).
Step 2. Calculate the next point (tk+1, xk+1, yk+1) using the formulas
tk+1 = tk + ∆t
and
xk+1 = xk + f(xk, yk) ∆t
and
yk+1 = yk + g(xk, yk) ∆t
Now, consider the second-order differential equation
d
2x
dt2
− (1 − x
2
)
dx
dt + x = 0. (6)
(FYI: Equation (6) is called the Van der Pol Equation.)
(OVER)
1Please review Section 1.4 – Numerical Technique: Euler’s Method.
(a) (2 points) By letting y = dx/dt, show that (6) can be converted into a first-order system
dx
dt = y
dy
dt = −x + (1 − x
2
)y.
(b) (8 points) Use Euler’s method for autonomous systems described above with the initial condition
x(0), y(0)
=
(x0, y0) = (1, 2) and step size ∆t = 0.1 to approximate
x(0.4), y(0.4)
. Create a table just like the one
below and fill it out. Also, only display the values for each entry and NOT the intermediate
calculations. Round your answers to four decimal places. (However, try to keep as many digits
as you can for intermediate calculations.)
k tk xk yk f(xk, yk) g(xk, yk)
0 0 1 2
1
2
3
4 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
x(0.4), y(0.4)
= ( , )
