University of Florida/Egm4313/s12.team4.Lorenzo/R1

From Wikiversity
Jump to navigation Jump to search

Report 1

Problem 6[edit]

Problem Statement[edit]

For each ODE in Fig.2 in K 2011 p.3 (except the last one involving a system of 2 ODEs), determine the order, linearity (or lack of), and show whether the principle of superposition can be applied. [1]

Theory[edit]

Order[edit]

The order of an equation is determined by the highest derivative. In this report, the first derivative of the y variable is denoted as y', and the second derivative is denoted as y, and so on. This can be determined upon observation of the equation.[2]

Linearity[edit]

An ordinary differential equation (ODE) is considered linear if it can be brought to the form[3]:

Superposition[edit]

Superposition can be applied if when a homogeneous and particular solution of an original equation are added, that they are equivalent to the original equation. In this report, variables with a bar over them represent the addition of the homogeneous and particular solution's same variable[4]. For example:


Given[edit]

The following equations were given in the textbook on p. 3[5]:

  • 1.6a -


  • 1.6b -


  • 1.6c -


  • 1.6d -


  • 1.6e -


  • 1.6f -


  • 1.6g -


  • 1.6h -


Solution[edit]

1.6a[edit]


order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution resembles the original equation, therefore superposition is possible


1.6b[edit]


order: 1st
linear: no
superposition: no

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible as also proven in the class notes [6]

1.6c[edit]


order: 1st
linear: no
superposition: no

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible

1.6d[edit]


order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution resembles the original equation, therefore superposition is possible


1.6e[edit]


order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution resembles the original equation, therefore superposition is possible


1.6f[edit]


order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution resembles the original equation, therefore superposition is possible


1.6g[edit]


order: 4th
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution resembles the original equation, therefore superposition is possible

1.6h[edit]


order: 2nd
linear: no
superposition:no

The given equation can be algebraically modified as the following:



It can be split up into the following homogeneous and particular solutions:



Adding the two solutions:



The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible


References[edit]

  1. Class Notes Iea.s12.sec2.djvu p. 2-4
  2. Kreyszig 2011 p.2
  3. Kreyszig 2011 p. 27
  4. [ http://www.coursesmart.com/SR/4279777/9780470458365/48 Kreyszig 2011 p. 48]
  5. Kreyszig 2011 p. 3
  6. Class Notes Iea.s12.sec2.djvu p. 2-4a (written example)

Reference Codes[edit]

[1]

[2]

[3]

[4]

[5]

[6]

  1. Kreyszig 2011 p.2
  2. Kreyszig 2011 p. 3
  3. Kreyszig 2011 p. 27
  4. [ http://www.coursesmart.com/SR/4279777/9780470458365/48 Kreyszig 2011 p. 48]
  5. Class Notes Iea.s12.sec2.djvu p. 2-4
  6. Class Notes Iea.s12.sec2.djvu p. 2-4a (written example)