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Massachusetts Institute of Technology

Department of Mechanical Engineering

2.087 - Engineering Mathematics: ODEs

Spring, 2017

Problem Set 1

Distributed:

Due:

Tuesday Feb 6, 2017

Tuesday Feb 14. 2017

Problem 1:

Below are several common differential equations. Label the order of each equation, and state whether it is

linear or nonlinear.

(a) (Spring equation) mx00 (t) + bx0 (t) + kx(t) = 0

(b) (Radioactive decay)

dQ

dt

= −rQ(t)

(c) (LRC circuit) LQ00 (t) + RQ0 (t) +

1

C Q(t)

= E(t)

(d) (Logistic equation) u0 (t) = λu(t)(1 − u(t))

(e) (Legendre differential equation) (1 − x2 )y 00 (x) − 2xy 0 (x) + `(` + 1)y(x) = 0

(f) (Compound interest) M 0 (t) = rM (t) + d

(g) (van der Pol equation)

d2 u

dt

+ (u2 − 1) du

dt + u = f (t)

Problem 2:

For each of the following ODEs, draw a direction field for the given differential equation. Based on the

direction field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at

t = 0, describe this dependency.

(a) y 0 = 3 − y

(b) y 0 = 2y − 5

(c) y 0 = y 2

(d) y 0 = −y(y − 3)

(e) y 0 = y(y − 2)2

Problem 3:

For each of the following, write down a differential equation of the form y 0 = ay + b whose solutions have

the required behavior as t → ∞.

(a) All solutions approach y = 3.

(b) All other solutions diverge from y = 1/3.

1

Problem 4:

For each of the following, draw a direction field for the given differential equation. Based on the direction

field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at t = 0,

describe this dependency.

(a) y 0 = −2 + t − y

(b) y 0 = te−2t − 2y

(c) y 0 = 3 sin t + 1 + y

Problem 5:

Show that the nonlinear example dy/dt = y 2 is solved by y = C/(1 − Ct) for every constant C.

Problem 6:

dy/dt = y + 1 is not solved by y = et + t. Substitute y to show that it fails. We can’t just add the solutions

to y 0 = y and y 0 = 1. What number c makes y = et + c into a correct solution?

Problem 7:

What linear differential equation y 0 = a(t)y is satisfied by y(t) = −esin t ?

2

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