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Massachusetts Institute of Technology
Department of Mechanical Engineering
2.087 - Engineering Mathematics: ODEs
Problem Set 1
Tuesday Feb 6, 2017
Tuesday Feb 14. 2017
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)
(c) (LRC circuit) LQ00 (t) + RQ0 (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)
+ (u2 − 1) du
dt + u = f (t)
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
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.
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
Show that the nonlinear example dy/dt = y 2 is solved by y = C/(1 − Ct) for every constant C.
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?
What linear differential equation y 0 = a(t)y is satisfied by y(t) = −esin t ?