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

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 21: Thurs, 8Oct09

Page 21-1

P.20-4 (continued)

$\Rightarrow a{x}^{-1}$ is a homgenous solution $\mathrm{\forall }a$

$u_1(x)=c_1{x}^{-1}$

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

$u_2(x)=c_2{x}^2$

(Verify $u_1$ and $u_2$ are linearly independant components of $W$

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

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

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

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

Page 21-2

Llinearity of ordinary differential equation $\Rightarrow$ superposition

$y_P(x)=\frac{1}{4}{x}^6+\frac{1}{6}{x}^5$

$y(x)=c_1{x}^{-1}+c_2{x}^2+y_P(x)$ , where $c_1{x}^{-1}+c_2{x}^2=y_H(x)$

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 $u_2$ and variation of parameter method)

Eq.(1) P.3-1 = $y^{″}+a_1(x)y^{\prime }+a_0(x)y=f(x)$

Assume having found $u_1(x)$, a homogeneous solution: ${u_1}^{″}+a_1(x){u_1}^{\prime }+a_0(x)u_1=0$

Consider: $y(x)=U(x)u_1(x)$ , where $U(x)$ is an undetermined factor

Page 21-3

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

$\begin{array}{c}f(x)=U^{\prime }(a_1{u}_1+2{u_1}^{\prime })+U^{″}{u}_1\end{array}$

NOTE: this equation is missing the dependant variable $U$ in front of $U^{\prime }$ term due to reduction of order method $\varphi$

$\begin{array}{c}Z(x):=U^{\prime }(x)\end{array}$

$\begin{array}{c}\Rightarrow u_1(x)Z^{\prime }+[a_1(x)u_1(x)+2{u_1}^{\prime }(x)]Z=f(x)\end{array}$

where $u_1(x)$ and $[a_1(x)u_1(x)+2{u_1}^{\prime }(x)]$ are known

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

$\begin{array}{c}U(x)=\underset{}{\overset{x}{\int }}Z(s)ds\end{array}$

$\begin{array}{c}y(x)=U(x)u_1(x)\end{array}$

ref: K p.28, problem 1.1ab

a) $u_1(x)=e^x$ , $(x-1)y^{″}-x{y}^{\prime }+y=0$

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

Page 21-4

Find $r_1,r_2$

How many valid homogeneous solutions to $u_1=e^{r_{1}x}$, find $u_2$ using undetermined factor method

References

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