User:Egm4313.s12.team11.Suh/R1.6

Problem Statement
 Problem 1.6  For each Ordinary Differential Equations, determine the order, linearity, and show whether the principal of superposition can be applied.

Order of ODEs
To find the order of Ordinary Differential Equations, (ODEs) simply check the differential order of the dependent variable. If the dependent variable is differentiated once, then it is a first order ODE. If it is differentiated twice, it is a second order ODE.

Linearity
To find the linearity of the ODE, check the power of the dependent variable. If the dependent variable is raised to the power of anything one than 1, then it is not linear. If the dependent variable is raised to the power of 1, then it is linear.

Superposition Principle
For a homogeneous linear ODE (2), any linear combination of two solutions on an open interval I is again a solution of (2) on I. In particular, for such an equation, sums and constant multiples of solutions are again solutions.  The function satisfies the superposition principle if the sum of the homogeneous solution and the particular solution is equal to $$\overline{y}(x)\!$$, with $$\overline{y}=(y_{h}+y_{p})\!$$ $$\overline{y}(x):=y_{h}(x)+y_{p}(x)\!$$

1.6a

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


 * Check superposition principle:
 * $$y''=g\!$$
 * $$y_{h}''=0\!$$
 * $$y_{p}''=g\!$$
 * $$(y_{h}+y_{p})\equiv\overline{y}''\!$$

Order of ODE: 2nd Order Linearity: Linear Superposition Principle: Yes

1.6b

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


 * Check superposition principle:
 * $$mv'+bv^2=mg\!$$
 * $$mv_{h}'+bv_{h}^2=0\!$$
 * $$mv_{p}'+bv_{p}^2=mg\!$$
 * $$m(v_{h}'+v_{p}')+b(v_{h}^2+v_{p}^2)\not\equiv m\overline{v}'+b(\overline{v}^2)\!$$

Order of ODE: 1st Order Linearity: Non-linear Superposition Principle: No

1.6c

 * $$h'=-k\sqrt{h} \!$$


 * Check superposition principle:
 * $$h'+kh^{1/2}=0\!$$
 * $$h_{h}'+kh_{h}^{1/2}=0\!$$
 * $$h_{p}'+kh_{p}^{1/2}=0\!$$
 * $$(h_{h}'+h_{p}')+k(h_{h}^{1/2}+h_{p}^{1/2})\not\equiv\overline{h}'+k\overline{h}^{1/2}\!$$

Order of ODE: 1st Order Linearity: Non-linear Superposition Principle: No

1.6d

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


 * Check superposition principle:
 * $$y''+\frac{k}{m}y=0\!$$
 * $$y_{h}''+\frac{k}{m}y_{h}=0\!$$
 * $$y_{p}''+\frac{k}{m}y_{p}=0\!$$
 * $$(y_{h}+y_{p})+\frac{k}{m}(y_{h}+y_{p})\equiv\overline{y}''+\frac{k}{m}\overline{y}\!$$

Order of ODE: 2nd Order Linearity: Linear Superposition Principle: Yes

1.6e

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


 * Check superposition principle:
 * $$y''+\omega _{o}^{2}y=\cos (\omega t)\!$$
 * $$y_{h}''+\omega _{o}^{2}y_{h}=0\!$$
 * $$y_{p}''+\omega _{o}^{2}y_{p}=\cos (\omega t)\!$$
 * $$(y_{h}+y_{p})+\omega _{o}^{2}(y_{h}+y_{p})\equiv\overline{y}''+\omega _{o}^{2}\overline{y}\!$$

Order of ODE: 2nd Order Linearity: Linear Superposition Principle: Yes

1.6f

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


 * Check superposition principle:
 * $$I''+\frac{R}{L}I'+\frac{1}{LC}I=\frac{E'}{L}\!$$
 * $$I_{h}''+\frac{R}{L}I_{h}'+\frac{1}{LC}I_{h}=0\!$$
 * $$I_{p}''+\frac{R}{L}I_{p}'+\frac{1}{LC}I_{p}=\frac{E'}{L}\!$$
 * $$(I_{h}+I_{p})+\frac{R}{L}(I_{h}'+I_{p}')+\frac{1}{LC}(I_{h}+I_{p})\equiv\overline{I}''+\frac{R}{L}\overline{I}'+\frac{1}{LC}\overline{I}\!$$

Order of ODE: 2nd Order Linearity: Linear Superposition Principle: Yes

1.6g

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


 * Check superposition principle:
 * $$y-\frac{1}{EI}y=0\!$$
 * $$y_{h}-\frac{1}{EI}y_{h}=0\!$$
 * $$y_{p}-\frac{1}{EI}y_{p}=0\!$$
 * $$(y_{h}'+y_{p}')-\frac{1}{EI}(y_{h}+y_{p})\equiv\overline{y}-\frac{1}{EI}\overline{y}\!$$

Order of ODE: 4th Order Linearity: Linear Superposition Principle: Yes

1.6h

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


 * Check superposition principle:
 * $$\theta''+\frac{g}{L}\sin\theta=0\!$$
 * $$\theta_{h}''+\frac{g}{L}\sin\theta_{h}=0\!$$
 * $$\theta_{p}''+\frac{g}{L}\sin\theta_{p}=0\!$$
 * $$(\theta_{h}+\theta_{p})+\frac{g}{L}(\sin\theta_{h}+\sin\theta_{p})\not\equiv\overline{\theta}''+\frac{g}{L}\sin\overline{\theta}\!$$

Order of ODE: 2nd Order Linearity: Non-linear Superposition Principle: No

1.6i

 * $$y_{1}^{'}=ay_{1}-by_{1}y_{2} \!$$
 * $$y_{2}^{'}=ky_{1}y_{2}-ly_{1} \!$$

Order of ODE: 1st Order Linearity: Non-linear Superposition Principle: N/A