# 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.

### 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.

#### Linearity

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

#### 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. For example:
$y_{p}+y_{h}={\overline {y}}$

### Given

The following equations were given in the textbook on p. 3:
* 1.6a - $y''=g=constant\!$
* 1.6b - $mv'=mg-bv^{2}\!$
* 1.6c - $h'=-k{\sqrt {h}}\!$
* 1.6d - $my''+ky=0\!$
* 1.6e - $y''+\omega _{0}^{2}y=\cos \omega t,\omega _{0}=\omega \!$
* 1.6f - $LI''+RI'+{\frac {1}{C}}I=E'\!$
* 1.6g - $EIy^{\omega }=f(x)\!$
* 1.6h - $L\theta ''+g\sin \theta =0\!$

### Solution
#### 1.6a

$y''=g=constant\!$
order: 2nd
linear: yes
superposition: yes
    The given equation can be algebraically modified as the following:
       $y''=g\!$
    It can be split up into the following homogeneous and particular solutions:
       $y_{h}''=0\!$
       $y_{p}''=g\!$
    Adding the two solutions:
       $(y_{h}''+y_{p}'')={\overline {y}}''\!$
    The solution resembles the original equation, therefore superposition is possible

#### 1.6b

$mv'=mg-bv^{2}\!$
order: 1st
linear: no
superposition: no
    The given equation can be algebraically modified as the following:
       $mv'+bv^{2}=mg\!$
    It can be split up into the following homogeneous and particular solutions:
       $mv_{h}'+bv_{h}^{2}=0\!$
       $mv_{p}'+bv_{p}^{2}=mg\!$
    Adding the two solutions:
       $m(v_{h}'+v_{p}')+b(v_{h}^{2}+v_{p}^{2})\not =m{\overline {v}}'+b({\overline {v}}^{2})\!$
    The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible as also proven in the class notes

### 1.6c

$h'=-k{\sqrt {h}}\!$
order: 1st
linear: no
superposition: no
    The given equation can be algebraically modified as the following:
       $h'+k{\sqrt {h}}=0\!$
    It can be split up into the following homogeneous and particular solutions:
       $h_{h}'+k{\sqrt {h_{h}}}=0\!$
       $h_{p}'+k{\sqrt {h_{p}}}=0\!$
    Adding the two solutions:
       $(h_{h}'+h_{p}')+k({\sqrt {h_{h}}}+{\sqrt {h_{p}}})\not ={\overline {h}}'+k{\sqrt {\overline {h}}}\!$
    The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible

#### 1.6d

$my''+ky=0\!$
order: 2nd
linear: yes
superposition: yes
    The given equation can be algebraically modified as the following:
       $y''+{\frac {k}{m}}y=0\!$
    It can be split up into the following homogeneous and particular solutions:
       $y_{h}''+{\frac {k}{m}}y_{h}=0\!$
       $y_{p}''+{\frac {k}{m}}y_{p}=0\!$
    Adding the two solutions:
       $(y_{h}''+y_{p}'')+{\frac {k}{m}}(y_{h}+y_{p})={\overline {y}}''+{\frac {k}{m}}{\overline {y}}\!$
    The solution resembles the original equation, therefore superposition is possible

#### 1.6e

$y''+\omega _{0}^{2}y=\cos \omega t,\omega _{0}=\omega \!$
order: 2nd
linear: yes
superposition: yes
    The given equation can be algebraically modified as the following:
       $y''+\omega _{o}^{2}y=\cos(\omega t)\!$
    It can be split up into the following homogeneous and particular solutions:
       $y_{h}''+\omega _{o}^{2}y_{h}=0\!$
       $y_{p}''+\omega _{o}^{2}y_{p}=\cos(\omega t)\!$
    Adding the two solutions:
       $(y_{h}''+y_{p}'')+\omega _{o}^{2}(y_{h}+y_{p})={\overline {y}}''+\omega _{o}^{2}{\overline {y}}\!$
    The solution resembles the original equation, therefore superposition is possible

#### 1.6f

$LI''+RI'+{\frac {1}{C}}I=E'\!$
order: 2nd
linear: yes
superposition: yes
    The given equation can be algebraically modified as the following:
       $I''+{\frac {R}{L}}I'+{\frac {1}{LC}}I={\frac {E'}{L}}\!$
    It can be split up into the following homogeneous and particular solutions:
       $I_{h}''+{\frac {R}{L}}I_{h}'+{\frac {1}{LC}}I_{h}=0\!$
       $I_{p}''+{\frac {R}{L}}I_{p}'+{\frac {1}{LC}}I_{p}={\frac {E'}{L}}\!$
    Adding the two solutions:
       $(I_{h}''+I_{p}'')+{\frac {R}{L}}(I_{h}'+I_{p}')+{\frac {1}{LC}}(I_{h}+I_{p})={\overline {I}}''+{\frac {R}{L}}{\overline {I}}'+{\frac {1}{LC}}{\overline {I}}\!$
    The solution resembles the original equation, therefore superposition is possible

#### 1.6g

$EIy''''=f(x)\!$
order: 4th
linear: yes
superposition: yes
    The given equation can be algebraically modified as the following:
       $y''''-{\frac {1}{EI}}y=0\!$
    It can be split up into the following homogeneous and particular solutions:
       $y_{h}''''-{\frac {1}{EI}}y_{h}=0\!$
       $y_{p}''''-{\frac {1}{EI}}y_{p}=0\!$
    Adding the two solutions:
       $(y_{h}''''+y_{p}'''')-{\frac {1}{EI}}(y_{h}+y_{p})={\overline {y}}''''-{\frac {1}{EI}}{\overline {y}}\!$
    The solution resembles the original equation, therefore superposition is possible

#### 1.6h

$L\theta ''+g\sin \theta =0\!$
order: 2nd
linear: no
superposition:no
    The given equation can be algebraically modified as the following:
       $\theta ''+{\frac {g}{L}}\sin \theta =0\!$
    It can be split up into the following homogeneous and particular solutions:
       $\theta _{h}''+{\frac {g}{L}}\sin \theta _{h}=0\!$
       $\theta _{p}''+{\frac {g}{L}}\sin \theta _{p}=0\!$
    Adding the two solutions:
       $(\theta _{h}''+\theta _{p}'')+{\frac {g}{L}}(\sin \theta _{h}+\sin \theta _{p})\not ={\overline {\theta }}''+{\frac {g}{L}}\sin {\overline {\theta }}\!$
    The solution cannot be algebraically modified to resemble the original equation, and therefore superposition is NOT possible