University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg21

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P.20-4 (continued)

${\displaystyle \Rightarrow ax^{-1}\ }$ is a homgenous solution ${\displaystyle \forall \!\,a\ }$

${\displaystyle u_{1}(x)=c_{1}x^{-1}\ }$

2) ${\displaystyle b=2\Rightarrow ax^{2}\ }$ is another homogeneous solution since ${\displaystyle b^{2}-b-2=0\ }$

${\displaystyle u_{2}(x)=c_{2}x^{2}\ }$

(Verify ${\displaystyle u_{1}\ }$ and ${\displaystyle u_{2}\ }$ are linearly independant components of ${\displaystyle W\ }$

3) ${\displaystyle b=6,a={\frac {1}{4}}\Rightarrow \ }$ left hand side of Eq(1) p20-4 ${\displaystyle =7x^{4}\ }$ , where ${\displaystyle =7x^{4}\ }$ is the 1st term on the right hand side

for ${\displaystyle b=6,a={\frac {1}{4}}\Rightarrow {\frac {1}{4}}x^{6}\ }$

4) ${\displaystyle b=5,a={\frac {1}{6}}\Rightarrow \ }$ left hand side of Eq(1) p20-4 ${\displaystyle =3x^{3}\ }$ , where ${\displaystyle =3x^{3}\ }$ is the 2nd term on the right hand side

for ${\displaystyle b=5,a={\frac {1}{6}}\Rightarrow {\frac {1}{6}}x^{5}\ }$

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Llinearity of ordinary differential equation ${\displaystyle \Rightarrow \ }$ superposition

${\displaystyle y_{P}(x)={\frac {1}{4}}x^{6}+{\frac {1}{6}}x^{5}\ }$

${\displaystyle y(x)=c_{1}x^{-1}+c_{2}x^{2}+y_{P}(x)\ }$ , where ${\displaystyle c_{1}x^{-1}+c_{2}x^{2}=y_{H}(x)\ }$

Alternative method to obtain full solution for non-homogeneous L2_ODE_VC knowing only one homogeneous solution (e.g. obtained by trial solution) (bypassing reduction of order method2-undertermined factor for ${\displaystyle u_{2}\ }$ and variation of parameter method)

Eq.(1) P.3-1 = ${\displaystyle y''+a_{1}(x)y'+a_{0}(x)y=f(x)\ }$

Assume having found ${\displaystyle u_{1}(x)\ }$, a homogeneous solution: ${\displaystyle u_{1}''+a_{1}(x)u_{1}'+a_{0}(x)u_{1}=0\ }$

Consider: ${\displaystyle y(x)=U(x)u_{1}(x)\ }$ , where ${\displaystyle U(x)\ }$ is an undetermined factor

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Follow the same argument as on P.17-2 to obtain:

${\displaystyle \displaystyle {\begin{aligned}f(x)=U'(a_{1}u_{1}+2u_{1}')+U''u_{1}\end{aligned}}}$	(1)

NOTE: this equation is missing the dependant variable ${\displaystyle U\ }$ in front of ${\displaystyle U'\ }$ term due to reduction of order method ${\displaystyle \phi \ \ }$

${\displaystyle \displaystyle {\begin{aligned}Z(x):=U'(x)\end{aligned}}}$	(2)

${\displaystyle \displaystyle {\begin{aligned}\Rightarrow u_{1}(x)Z'+\left[a_{1}(x)u_{1}(x)+2u_{1}'(x)\right]Z=f(x)\end{aligned}}}$	(3)

where ${\displaystyle u_{1}(x)\ }$ and ${\displaystyle \left[a_{1}(x)u_{1}(x)+2u_{1}'(x)\right]\ }$ are known

Non-homogeneous L1_ODE_VC solution for ${\displaystyle Z(x)\ }$ : Eq.(4) P.8-2

${\displaystyle \displaystyle {\begin{aligned}U(x)=\int _{}^{x}Z(s)\,ds\end{aligned}}}$	(4)

${\displaystyle \displaystyle {\begin{aligned}y(x)=U(x)u_{1}(x)\end{aligned}}}$	(5)

ref: K p.28, problem 1.1ab

a) ${\displaystyle u_{1}(x)=e^{x}\ }$ , ${\displaystyle (x-1)y''-xy'+y=0\ }$

Trial solution ${\displaystyle y(x)=e^{rx}\ }$ , where ${\displaystyle r=\ }$ constant

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Find ${\displaystyle r_{1},r_{2}\ }$

How many valid homogeneous solutions to ${\displaystyle u_{1}=e^{r_{1}x}\ }$, find ${\displaystyle u_{2}\ }$ using undetermined factor method

References