University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg17

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Linearity ${\displaystyle \Rightarrow \ \ }$ superposition ${\displaystyle y=y_{H}+y_{P}\ }$

Homogeneous Solution ${\displaystyle y_{H}\ }$
- Euler Equations
- Trial solution (undefined coefficient)
- Reduction of order method 2: Undetermined factor

Homogeneous L2_ODE_VC: cf. Eq.(1) P.3-1

${\displaystyle \displaystyle {\begin{aligned}y''+a_{1}(x)y'+a_{0}y=0\end{aligned}}}$	(1)

Where ${\displaystyle a_{0}(x)\ }$ can be substituted for ${\displaystyle a_{1}(x)\ }$ or ${\displaystyle a_{0}y\ }$ in Eq(1)

Given one homogeneous solution ${\displaystyle u_{1}(x)\ }$ known

Find second homogeneous solution ${\displaystyle u_{2}(x)\ }$ such that

${\displaystyle \displaystyle {\begin{aligned}y_{H}(x)=k_{1}u_{1}(x)+k_{2}u_{2}(x)\end{aligned}}}$	(3)

Where${\displaystyle k_{1},k_{2}\ }$ are constants

Assume full homogeneous solution

${\displaystyle \displaystyle {\begin{aligned}y(x)=U(x)u_{1}(x)\end{aligned}}}$	(2)

Where${\displaystyle U(x)\ }$ is an unknown to be determined

Where${\displaystyle u_{1}(x)\ }$ is known

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"Full" = includes ${\displaystyle u_{2}(x)\ }$

Add the following: ${\displaystyle a_{0}(x)\left[y=Uu_{1}\right]\ }$

and ${\displaystyle a_{1}(x)\left[y'=Uu_{1}'+U'u_{1}\right]}$

and ${\displaystyle \left[y''=Uu_{1}''+2U'u_{1}'+U''u_{1}\right]}$

To get ${\displaystyle a_{0}y+a_{1}y'+y''=U\left[a_{0}u_{1}+a_{1}u_{1}'+u_{1}''\right]+U'\left[a_{1}u_{1}+2u_{1}'\right]+U''u_{1}=0\ }$ by Eq(1) p17-1

Reduce to ${\displaystyle u_{1}''+a_{1}u_{1}'+a_{0}u_{1}=0\ }$

Since ${\displaystyle u_{1}\ }$ is a homogeneous solution ${\displaystyle \Rightarrow \ 0=U'(a_{1}u_{1}+2u_{1}')+U''u_{1}\ }$, NOTE missing dependent variable U in front of ${\displaystyle U'\ }$ term

Let ${\displaystyle Z:=U'\Rightarrow \ \ }$ homogeneous L1_ODE_VC for Z

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${\displaystyle \displaystyle {\begin{aligned}\Rightarrow \ u_{1}(x)Z'+(a_{1}(x)u_{1}(x)+2u_{1}'(x)Z=0\end{aligned}}}$	(1)

Solve for Z,
- integration factorial method (HW)
- Direct integration (because Eq(1) is homogeneous)

${\displaystyle \Rightarrow \ {\frac {Z'}{Z}}+(a_{1}+{\frac {2u_{1}'}{u_{1}}})=0\ }$

Where ${\displaystyle a_{1},{\frac {2u_{1}'}{u_{1}}}\ }$ are known

Integrate ${\displaystyle \log \left|Z\right\vert +2\log \left|u_{1}\right\vert +\int _{}^{x}a_{1}(s)\,ds=k}$, where k is a constant

${\displaystyle \displaystyle {\begin{aligned}Z(x)={\frac {c}{(u_{1})^{2}}}e^{-\int _{}^{x}a_{1}(s)\,ds}=U'(x)\end{aligned}}}$	(2)

where ${\displaystyle c=e^{k}\ }$

${\displaystyle \Rightarrow \ U(x)=\int _{}^{x}{\frac {c}{(u_{1}(t))^{2}}}e^{-\int _{}^{t}a_{1}(s)\,ds}\,dt+{\tilde {c}}\ }$

where ${\displaystyle {\bar {a}}\ _{1}(t)=\int _{}^{t}a_{1}(s)\,ds\ }$

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Homogeneous solution

${\displaystyle \displaystyle {\begin{aligned}y(x)=U(x)u_{1}(x)=cu_{1}\int _{}^{x}{\frac {1}{(u_{1}(t))^{2}}}e^{-{\bar {a}}\ _{1}(t)}\,dt+{\tilde {c}}u_{1}={\tilde {c}}u_{1}+cu_{2}\end{aligned}}}$	(1)

where ${\displaystyle {\tilde {c}}=k_{1}\ }$ and ${\displaystyle c=k_{2}\ }$

${\displaystyle \displaystyle {\begin{aligned}\Rightarrow u_{2}=u_{1}\int _{}^{x}{\frac {1}{(u_{1}(t))^{2}}}e^{-{\bar {a}}\ _{1}(t)}\,dt\end{aligned}}}$	(2)

HW: obtain Eq.(2) P.17-3 ${\displaystyle Z(x)\ }$ using the integrating factor method

References