# User:Egm4313.s12.team6.berthoumieux

## Contents

### 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 wether the principle of superposition can be applied.

### Given

$(1){y}''=g=constant$ $(2)m{v}'=mg-bv^{2}$ $(3){h}'=-k{\sqrt {h}}$ $(4)m{y}''+ky=0$ $(5)y''+\omega _{0}^{2}=cos(\omega t)$ , $\omega _{0}=\omega$ $(6)L{I}''+R{I}'+{\frac {1}{c}}I={E}'$ $(7)EIy^{iv}={\mathit {f(x)}}$ $(8){\mathit {L{\theta }''+gsin{\theta }=0}}$ ### Solution

$(1){y}''=g=constant$ Order:2nd order. The highest derivative is a 2nd derivative on the y.
Linearity:Linear.
Superposition:Yes.

$(2)m{v}'=mg-bv^{2}$ Order:1st order. The highest derivative is a 1st on the v.
Linearity:Non-linear.
Superposition:No.

$(3){h}'=-k{\sqrt {h}}$ Order:1st order
Linearity:Non-linear.
Superposition:No.

$(4)m{y}''+ky=0$ Order:2nd order.
Linearity:Linear.
Superposition:Yes.

$(5)y''+\omega _{0}^{2}=cos(\omega t)$ , $\omega _{0}=\omega$ Order:2nd order
Linearity:Linear.
Superposition:Yes.

$(6)L{I}''+R{I}'+{\frac {1}{c}}I={E}'$ Order:2nd order
Linearity:Linear.
Superposition:Yes.

$(7)EIy^{iv}={\mathit {f(x)}}$ Order:4th oder
Linearity:Linear.
Superposition:Yes.

$(8){\mathit {L{\theta }''+gsin{\theta }=0}}$ Order:2nd order
Linearity:Non-linear.
Superposition:Yes.

### Statement

Realize spring-mass-dashpot systems in series as shown in the figure.

With characteristic: ${\lambda }^{2}-10{\lambda }+25=0$ And with double real root at ${\lambda }=-3$ Find the values of k, c, m.

### Solution

First, we write down the equations we can figure out from kinematics and from kinetics.
Kinematics: $y=y_{k}+y_{c}$ Kinetics:$m{y}''+f_{l}=f(t)$ $f_{l}=f_{k}=f_{c}$ Next, we can find any relations that exist:
$f_{k}=ky_{k}$ $f_{c}=c{y_{c}}'$ Since we already established that $y=y_{k}+y_{c}$ , we can say that ${y}''={y_{k}}''+{y_{c}}''$ We also established that $f_{k}=f_{c}$ so we can say that $ky_{y}=c{y_{c}}'n\to {y_{c}}'={\frac {k}{c}}y_{k}$ Next, we plug in the value for ${y_{c}}'$ :
${y}''={y_{k}}''+{({y_{c}}')}'={y_{k}}''+{\frac {k}{c}}{y_{k}}'$ Finally, plug ${y}''$ back into the kinetics equation:
$m({y_{k}}''+{\frac {k}{c}}{y_{k}}')+ky_{k}=f(t)$ $m{y_{k}}''+m{\frac {k}{c}}{y_{k}}'+ky_{k}=f(t)$ We are given that ${\lambda }=-3$ so the characteristic equation must be:
$(\lambda -(-3))^{2}=0$ $\lambda ^{2}+6\lambda +9=0$ This relates to the second order ODE:
${y}''+6{y}'+9y=0$ Comparing this to the following equation:
$m{y_{k}}''+m{\frac {k}{c}}{y_{k}}'+ky_{k}=f(t)$ We find that

      $m=1$ $k=9$ $c={\frac {9}{6}}={\frac {3}{2}}$ 