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

Report 1

## Problem 6

### Problem Statement

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

#### Order

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

An ordinary differential equation (ODE) is considered linear if it can be brought to the form[3]: ${\displaystyle y'+p(x)y=q(x)\!}$

#### Superposition

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:

${\displaystyle y_{p}+y_{h}={\overline {y}}}$

### Given

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

• 1.6a - ${\displaystyle y''=g=constant\!}$

• 1.6b - ${\displaystyle mv'=mg-bv^{2}\!}$

• 1.6c - ${\displaystyle h'=-k{\sqrt {h}}\!}$

• 1.6d - ${\displaystyle my''+ky=0\!}$

• 1.6e - ${\displaystyle y''+\omega _{0}^{2}y=\cos \omega t,\omega _{0}=\omega \!}$

• 1.6f - ${\displaystyle LI''+RI'+{\frac {1}{C}}I=E'\!}$

• 1.6g - ${\displaystyle EIy^{\omega }=f(x)\!}$

• 1.6h - ${\displaystyle L\theta ''+g\sin \theta =0\!}$

### Solution

#### 1.6a

${\displaystyle y''=g=constant\!}$
order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:

${\displaystyle y''=g\!}$

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

${\displaystyle y_{h}''=0\!}$
${\displaystyle y_{p}''=g\!}$

${\displaystyle (y_{h}''+y_{p}'')={\overline {y}}''\!}$

The solution resembles the original equation, therefore superposition is possible

#### 1.6b

${\displaystyle mv'=mg-bv^{2}\!}$
order: 1st
linear: no
superposition: no

The given equation can be algebraically modified as the following:

${\displaystyle mv'+bv^{2}=mg\!}$

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

${\displaystyle mv_{h}'+bv_{h}^{2}=0\!}$
${\displaystyle mv_{p}'+bv_{p}^{2}=mg\!}$

${\displaystyle m(v_{h}'+v_{p}')+b(v_{h}^{2}+v_{p}^{2})\not =m{\overline {v}}'+b({\overline {v}}^{2})\!}$

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

${\displaystyle h'=-k{\sqrt {h}}\!}$
order: 1st
linear: no
superposition: no

The given equation can be algebraically modified as the following:

${\displaystyle h'+k{\sqrt {h}}=0\!}$

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

${\displaystyle h_{h}'+k{\sqrt {h_{h}}}=0\!}$
${\displaystyle h_{p}'+k{\sqrt {h_{p}}}=0\!}$

${\displaystyle (h_{h}'+h_{p}')+k({\sqrt {h_{h}}}+{\sqrt {h_{p}}})\not ={\overline {h}}'+k{\sqrt {\overline {h}}}\!}$

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

#### 1.6d

${\displaystyle my''+ky=0\!}$
order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:

${\displaystyle y''+{\frac {k}{m}}y=0\!}$

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

${\displaystyle y_{h}''+{\frac {k}{m}}y_{h}=0\!}$
${\displaystyle y_{p}''+{\frac {k}{m}}y_{p}=0\!}$

${\displaystyle (y_{h}''+y_{p}'')+{\frac {k}{m}}(y_{h}+y_{p})={\overline {y}}''+{\frac {k}{m}}{\overline {y}}\!}$

The solution resembles the original equation, therefore superposition is possible

#### 1.6e

${\displaystyle y''+\omega _{0}^{2}y=\cos \omega t,\omega _{0}=\omega \!}$
order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:

${\displaystyle y''+\omega _{o}^{2}y=\cos(\omega t)\!}$

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

${\displaystyle y_{h}''+\omega _{o}^{2}y_{h}=0\!}$
${\displaystyle y_{p}''+\omega _{o}^{2}y_{p}=\cos(\omega t)\!}$

${\displaystyle (y_{h}''+y_{p}'')+\omega _{o}^{2}(y_{h}+y_{p})={\overline {y}}''+\omega _{o}^{2}{\overline {y}}\!}$

The solution resembles the original equation, therefore superposition is possible

#### 1.6f

${\displaystyle LI''+RI'+{\frac {1}{C}}I=E'\!}$
order: 2nd
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:

${\displaystyle I''+{\frac {R}{L}}I'+{\frac {1}{LC}}I={\frac {E'}{L}}\!}$

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

${\displaystyle I_{h}''+{\frac {R}{L}}I_{h}'+{\frac {1}{LC}}I_{h}=0\!}$
${\displaystyle I_{p}''+{\frac {R}{L}}I_{p}'+{\frac {1}{LC}}I_{p}={\frac {E'}{L}}\!}$

${\displaystyle (I_{h}''+I_{p}'')+{\frac {R}{L}}(I_{h}'+I_{p}')+{\frac {1}{LC}}(I_{h}+I_{p})={\overline {I}}''+{\frac {R}{L}}{\overline {I}}'+{\frac {1}{LC}}{\overline {I}}\!}$

The solution resembles the original equation, therefore superposition is possible

#### 1.6g

${\displaystyle EIy''''=f(x)\!}$
order: 4th
linear: yes
superposition: yes

The given equation can be algebraically modified as the following:

${\displaystyle y''''-{\frac {1}{EI}}y=0\!}$

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

${\displaystyle y_{h}''''-{\frac {1}{EI}}y_{h}=0\!}$
${\displaystyle y_{p}''''-{\frac {1}{EI}}y_{p}=0\!}$

${\displaystyle (y_{h}''''+y_{p}'''')-{\frac {1}{EI}}(y_{h}+y_{p})={\overline {y}}''''-{\frac {1}{EI}}{\overline {y}}\!}$

The solution resembles the original equation, therefore superposition is possible

#### 1.6h

${\displaystyle L\theta ''+g\sin \theta =0\!}$
order: 2nd
linear: no
superposition:no

The given equation can be algebraically modified as the following:

${\displaystyle \theta ''+{\frac {g}{L}}\sin \theta =0\!}$

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

${\displaystyle \theta _{h}''+{\frac {g}{L}}\sin \theta _{h}=0\!}$
${\displaystyle \theta _{p}''+{\frac {g}{L}}\sin \theta _{p}=0\!}$

${\displaystyle (\theta _{h}''+\theta _{p}'')+{\frac {g}{L}}(\sin \theta _{h}+\sin \theta _{p})\not ={\overline {\theta }}''+{\frac {g}{L}}\sin {\overline {\theta }}\!}$

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

## References

### Reference Codes

[2]