18.03SC Differential Equations Fall 2011

发布时间：2023-09-03

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

18.03sc unit 1 Exam

1. (a) In a perfect environment, the population of Norway rat that breeds on the MIT campus increases by a factor of e = 2.718281828459045 . . . each year. Model this natural growth by a differential equation. [8]

What is the growth rate k?

(b) MIT is a limited environment, with a maximal sustainable Norway rat population of R = 1000 rats. write down the logistic equation modeling this. (you may use “k” for the natural growth rate here if you failed to ind it in (a).) [4]

(c) The MIT pest control service intends to control these rats by killing themat a constant rate of a rats per year. If it wants to limit the rat population to 75% of the maximal sustain-able population, what rate a it should aim for (in rats per year)? [8]

2. For the autonomous equation x(.) = x (x - 1)(x + 2), please sketch:

(a) the phase line, identifying the critical points and whether they are stable, unstable, or neither. [4]

(b) at least one solution of each basic type (so that every solution is a time-translate of one you have drawn) [4]

Below is a diagram of a direction ield of the differential equation y\ = (1/4)(x - y2 ). On it please plot and label:

(c) the nullcline [3]

(d) at least two quite different solutions [3]

(e) theseparatrix (if there is one) [3]

(f) True or false: If y(x) is a solution with a minimum, then for all large enough x, y(x) < [3] ^x. (No explanation needed: just circle one.)

3. (a) use Euler,s method with stepsize h = 1/2 to estimate the value at x = 3/2 of the [10] solution toy、= x + y such y(0) = 1.

(b) Find the solution of tx(.) + x = cos t such that x (π) = 1. [10]

4. (a) Find real u, b such that = u + bi. [3]

(b) Find real r, θ such that 1 - i = reiθ . [3]

(c) Find real u, b such that (1 - i)8 = u + bi. [3]

(d) Find real u, b such that b > 0 and u + bi is a cube root of -1. [3]

(e) Find real u, b such that eln 2+iπ = u + bi. [3]

(f) Write f (t) = 2 cos(4t) - 2 sin(4t) in the form A cos(“t - Φ). [5]

5. (a) Find a particular solution to the equation x(.) + 3x = e2t. [5]

(b) Find the solution to the same equation such that x (0) = 1. [5]

(c) Write down a linear equation with exponential right hand side of which [5]

x(.) + 3x = cos (2t) is the real part.

(d) Find a particular solution to the equation x(.) + 3x = cos (2t). [5]