University of Florida/Egm4313/s12.team8.dupre/R5.4

Problem Statement

(1) Show that:

$y_p(x)=\sum _{i=0}^n{y}_{p,i}(x)$ (5.1)

is indeed the overall particular solution of the L2-ODE-VC:

$y^{″}+p(x)y^{\prime }+q(x)y=r(x)$ (5.2)

with the excitation:

$r(x)=r_1(x)+r_2(x)+...+r_n(x)=\sum _{i=0}^n{r}_i(x)$ (5.3)

(2) Discuss the choice of $y_p(x)$ in the above table , e.g., for:

$r(x)=kcos(\omega x)$

Why would you need to have both $cos(\omega x),sin(\omega x)$ in $y_p(x)$?

Solution (1)

Using the following equation:

$r_i(x)=y_{p^{″},i}+p(x)y_{p^{\prime },i}+q(x)y_{p,i}$ (5.4)

for different r and y values gives us the following:

$r_1(x)=y_{p^{″},1}+p(x)y_{p^{\prime },1}+q(x)y_{p,1}$ (5.5)

$r_2(x)=y_{p^{″},2}+p(x)y_{p^{\prime },2}+q(x)y_{p,2}$ (5.6)

$r_3(x)=y_{p^{″},3}+p(x)y_{p^{\prime },3}+q(x)y_{p,3}$ (5.7)

Now, adding (5.4),(5.5), and (5.6), gives us:

$r_1(x)+r_2(x)+r_3(x)=(y_{p,1}+y_{p,2}+y_{p,3}{)}^{″}+p(x)(y_{p,1}+y_{p,2}+y_{p,3}{)}^{\prime }+q(x)(y_{p,1}+y_{p,2}+y_{p,3})$ (5.8)

Equation (5.8) shows us that the overall particular solution of (5.2) with excitation (5.3), is in fact, equation (5.1).

Solution (2)

We know that the given example for an excitation is the periodic excitation:

$r(x)=kcos(\omega x)$

When we decompose a periodic excitation into a Fourier trigonometric series, we find:

$r(x)=a_0+\sum _{n=0}^{\mathrm{\infty }}[a_{n}cos(n\omega x)+b_{n}sin(\omega x)]$

Since we know that the particular solution should depend on the excitation, we know that for a periodic excitation $r(x)$, we would need both $cos(\omega x),sin(\omega x)$ in $y_p(x)$ to obtain the correct particular solution.