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)

= ( , )