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Analytical Methods for Chem. & Biochem. Eng. Homework Set 2 Solutions, Fall 2009, Assignments of Engineering

Rutgers University - CamdenEngineering

The solutions to homework set 2 in the course 'analytical methods for chemical and biochemical engineering' taught by prof. Marianthi ierapetritou in the fall 2009 semester. Solutions to various differential equation problems, ranging from finding critical points to solving specific equations using different methods.

Typology: Assignments

Pre 2010

Uploaded on 10/31/2009

raygchen
raygchen 🇺🇸

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155:507 Analytical Methods for Chemical and Biochemical Engineering
Fall 2009: Homework Set 2 Solutions
Instructor: Prof. Marianthi Ierapetritou
Assigned 09/09 Due: 09/16
1. (2-Rev:5) The number 0 is a critical point of the autonomous differential equation n
x
dt
dx =,
when n is a positive integer. For what values of n is 0 asymptotically stable? Semistable?
Unstable? Repeat for the equation n
x
dt
dx −= .
2. (2-rev:8) Classify each differential equation as to: separable, exact, linear, homogeneous, or
Bernoulli. Some equations may be more than one kind. Do not solve but justify your answer.
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155:507 Analytical Methods for Chemical and Biochemical Engineering Fall 2009: Homework Set 2 Solutions Instructor: Prof. Marianthi Ierapetritou Assigned 09/09 Due: 09/

  1. (2-Rev:5) The number 0 is a critical point of the autonomous differential equation xn dt

dx = ,

when n is a positive integer. For what values of n is 0 asymptotically stable? Semistable? Unstable? Repeat for the equation xn dt

dx = −.

  1. (2-rev:8) Classify each differential equation as to: separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve but justify your answer.
  1. (2-Rev:10) Solve the following differential equation: y (ln x−lny)dx=(xlnx−xlny−y) dy
  2. (2-Rev:14) Solve the following differential equation:

( 2 x+ y+ 1 )y'^ = 1

  1. (2.5:37) For the following differential equation:

xv dv v 2 32 x dx

  • =

(a) solve the equation by converting it into an exact equation using an integrating factor

(b) solve the DE using the fact that it is a Bernoulli equation

  1. (2.5:27) Solve the following DE using the appropriate substitution

2 2 3 dy y x dx

= + − +

  1. (2.5:13) Solve the following initial value problem

( x + ye y x^ /^ ) dx − xe y x/ dy = 0, y(1) = 0

  1. (3.1:39) (a) Verify that y 1 = x^3 and 3 y 2 = x are linearly independent solutions of the

differential equation x 2 y′′^ − 4 xy′+ 6 y= 0 on the interval ( −∞, ∞).

(b) Show that W (y 1 ,y 2 )= 0 for every real number x. Does this result violates theorem 3.3? Explain.