User:Egm4313.s12.team2.dzadek.rl

About Me
My name is Rachel Dzadek. I am a mechanical engineering student at the University of Florida. I am in Team 2

R1.6
QUESTION: 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.

Falling Stone

 * $$ y'' = g = const. \ $$

Order: 2nd order Linearity (or lack of): linear Superposition:
 * $$ y_h '' = 0                       \ $$
 * $$ y_p '' = g                       \ $$
 * $$ y(x) := (y_h + y_p)          \ $$


 * $$ y(x) = y_h  + y_p '' = 0 + g \ $$
 * $$ y(x) = (y_p + y_h)           \ $$
 * $$ y(x)'' = g                       \ $$

Superposition true $$ \therefore $$ linear.

Parachute

 * $$ mv' = mg - bv^2 \ $$

Order: 1st order Linearity (or lack of): not linear Superposition:
 * $$ y_h = mv_h ' + kv_h ^2 = 0                 \ $$
 * $$ y_p = mv_p ' + kv_p ^2 = mg                \ $$
 * $$ y(x) := y_h + y_p                          \ $$


 * $$ y(x) = mv_h ' + kv_h ^2 + mv_p ' + kv_p ^2 \ $$
 * $$ y_h + y_p = 0 + mg                         \ $$


 * $$ mv_h ' + kv_h ^2 + mv_p ' + kv_p ^2 = mg   \ $$
 * $$ m(v_h ' + v_p ') + k(v_h ^2 + v_p ^2) = mg \ $$
 * $$ m(v_h + v_p)' + k(v_h + v_p)^2 \neq mg     \ $$

Superposition false $$ \therefore $$ not linear.

Outflowing Water

 * $$ h' = -k \sqrt(h) \ $$

Order: 1st order Linearity (or lack of): not linear Superposition :
 * $$ y_h = h_h ' + k \sqrt h_h = 0         \ $$
 * $$ y_p = h_p ' = -k \sqrt h_p            \ $$
 * $$ y(x) := y_h + y_p                     \ $$


 * $$ y(x) = h_h ' + k \sqrt h_h + h_p '    \ $$
 * $$ y_h + y_p = 0 + -k \sqrt h_p          \ $$


 * $$ \frac {dh} {dt} = -kh^ \frac 1 2      \ $$
 * $$ \int h^ {- \frac 1 2} dh = - \int k dt \ $$
 * $$ 2h^ \frac 1 2 = -kt + c               \ $$
 * $$ h^ \frac 1 2 = - \frac {kt} 2 + c     \ $$
 * $$ h = ( - \frac {kt} 2 + c )^2          \ $$


 * $$ h(1) = ( - \frac k 2 + c )^2          \ $$
 * $$ h(2) = ( -k + c )^2                   \ $$
 * $$ h(3) = ( - \frac {3k} 2 + c )^2       \ $$
 * $$ h(1) \neq h(2) \neq h(3)              \ $$

Superposition false $$ \therefore $$ not linear.

Vibrating Mass on a Spring

 * $$ my'' + ky = 0 \ $$

Order: 2nd order Linearity (or lack of): linear Superposition :
 * $$ my'' + ky = 0                                                      \ $$
 * $$ y'' = - \frac k m y                                                \ $$

Let $$ y_1 \ $$ and $$ y_2 \ $$ be solutions.
 * $$ y_1 '' = - \frac k m y_1                                           \ $$
 * $$ y_2 '' = - \frac k m y_2                                           \ $$


 * $$ (ay_1 + by_2)'' = - \frac k m (ay_1 + by_2)                        \ $$
 * $$ ay_1  + by_2  = - \frac k m (ay_1 + by_2)                      \ $$
 * $$ a(- \frac k m y_1) + b(- \frac k m y_2) = - \frac k m (ay_1 + by_2) \ $$
 * $$ - \frac k m (ay_1 + by_2) = - \frac k m (ay_1 + by_2)              \ $$

Superposition true $$ \therefore $$ linear.

Beats of a Vibrating System

 * $$ y'' + \omega _0 ^2 y = \cos \omega t \ $$
 * $$ \omega_0 = \omega \ $$

Order: 2nd order Linearity (or lack of): linear Superposition:
 * $$ y_h = y_h '' + \omega _0 ^2 y_h = 0                                  \ $$
 * $$ y_p = y_p '' + \omega _0 ^2 y_p = \cos \omega t                      \ $$
 * $$ y(x) := y_h + y_p                                                    \ $$


 * $$ y(x) = y_h  + \omega _0 ^2 y_h + y_p  + \omega _0 ^2 y_p         \ $$
 * $$ y_h + y_p = 0 + \cos \omega t                                        \ $$


 * $$ y_h  + \omega _0 ^2 y_h + y_p  + \omega _0 ^2 y_p = \cos \omega t \ $$
 * $$ (y_h  + y_p ) + \omega _0 ^2 (y_h + y_p) = \cos \omega t         \ $$
 * $$ (y_h + y_p)'' + \omega _0 ^2 (y_h + y_p) = \cos \omega t             \ $$
 * $$ y'' + \omega _0 ^2 y = \cos \omega t                                 \ $$

Superposition true $$ \therefore $$ linear.

Current I in an RLC Circuit

 * $$ LI'' + RI' + \frac 1 C I = E' \ $$

Order: 2nd order Linearity (or lack of): linear Superposition:
 * $$ y_h = LI_h '' + RI_h ' + \frac 1 C I_h = 0                                        \ $$
 * $$ y_p = LI_p '' + RI_p ' + \frac 1 C I_p = E'                                       \ $$
 * $$ y(x) := y_h + y_p                                                                 \ $$


 * $$ y(x) = LI_h  + RI_h ' + \frac 1 C I_h + LI_p  + RI_p ' + \frac 1 C I_p        \ $$
 * $$ y_h + y_p = 0 + E'                                                                \ $$


 * $$ LI_h  + RI_h ' + \frac 1 C I_h + LI_p  + RI_p ' + \frac 1 C I_p = E'          \ $$
 * $$ L(I_h  + I_p ) + R(I_h ' + I_p ') + \frac 1 C (I_h + I_p) = E'                \ $$
 * $$ L(I_h + I_p)'' + R(I_h + I_p)' + \frac 1 C (I_h + I_p) = E'                       \ $$
 * $$ LI'' + RI' + \frac 1 C I = E'                                                     \ $$

Superposition true $$ \therefore $$ linear.

Deformation of a Beam

 * $$ EIy^{IX} = f(x) \ $$

Order: 4th order Linearity (or lack of): linear Superposition:
 * $$ y_h = EIy_h ^{IX} = 0           \ $$
 * $$ y_p = EIy_p ^{IX} = f(x)        \ $$
 * $$ y(x) := y_h + y_p               \ $$


 * $$ y(x) = EIy_h ^{IX} + EIy_p ^{IX} \ $$
 * $$ y_h + y_p = 0 + f(x)            \ $$


 * $$ EIy_h ^{IX} + EIy_p ^{IX} = f(x) \ $$
 * $$ EI(y_h ^{IX} + y_p ^{IX}) = f(x) \ $$
 * $$ EI(y_h + y_p)^{IX} = f(x)       \ $$
 * $$ EIy^{IX} = f(x)                 \ $$

Superposition true $$ \therefore $$ linear.

Pendulum

 * $$ L \theta '' + g \sin \theta = 0 \ $$

Order: 2nd Linearity (or lack of): not linear Superposition:
 * $$ y_h = L \theta _h '' + g \sin \theta_h = 0                                      \ $$
 * $$ y_p = L \theta _p '' + g \sin \theta_p = 0                                      \ $$
 * $$ y(x) := y_h + y_p                                                               \ $$


 * $$ y(x) = L \theta _h  + g \sin \theta_h + L \theta _p  + g \sin \theta_p      \ $$
 * $$ y_h + y_p = 0 + 0                                                               \ $$


 * $$ L \theta _h  + g \sin \theta_h + L \theta _p  + g \sin \theta_p = 0         \ $$
 * $$ L( \theta _h  + \theta _p ) + g( \sin \theta_h + \sin \theta_p) = 0         \ $$
 * $$ L( \theta _h + \theta _p)'' + g( \sin \theta_h + \sin \theta_p) = 0             \ $$
 * $$ L \theta  + g( \sin \theta_h + \sin \theta_p) \neq L \theta  + g \sin \theta \ $$

Superposition false $$ \therefore $$ not linear.