University of Florida/Egm4313/s12.team11.gooding/R5

Problem 5.5

Part 1

Problem Statement

Show that $cos(7x)$ and $sin(7x)$ are linearly independant using the Wronskian and the Gramain (integrate over 1 period)

Solution

$f=cos(7x),g=sin(7x)$
One period of $7x=\pi /7$
Wronskian of f and g
$W(f,g)=det{\begin{bmatrix}f&g\\f'&g'\end{bmatrix}}$

Plugging in values for $f,f',g,g';$
$W(f,g)=det{\begin{bmatrix}cos(7x)&sin(7x)\\-sin(7x)&cos(7x)\end{bmatrix}}$ $=7cos^{2}(7x)+7sin^{2}(7x)$
$=7[cos^{2}(7x)+sin^{2}(7x)]$
$=7[1]$

 They are linearly Independant using the Wronskian.

$<f,g>=\int _{a}^{b}f(x)g(x)dx$
$\Gamma (f,g)=det{\begin{bmatrix}<f,f>&<f,g>\\<g,f>&<g,g>\end{bmatrix}}$
$\int _{0}^{\pi /7}cos^{2}(7x)dx=\pi /14$
$\int _{0}^{\pi /7}sin^{2}(7x)dx=\pi /14$
$\int _{0}^{\pi /7}cos(7x)*sin(7x)dx=0$
$\Gamma (f,g)=det{\begin{bmatrix}\pi /14&0\\0&\pi /14\end{bmatrix}}$
$\Gamma (f,g)=\pi ^{2}/49$

 They are linearly Independent using the Gramain.

Problem Statement

Find 2 equations for the 2 unknowns M,N and solve for M,N.

Solution

$y_{p}(x)=Mcos7x+Nsin7x$
$y'_{p}(x)=-M7sin7x+N7cos7x$
$y''_{p}(x)=-M7^{2}cos7x-N7^{2}sin7x$
Plugging these values into the equation given ($y''-3y'-10y=3cos7x$) yields;
$-M7^{2}cos7x-N7^{2}sin7x-3(-M7sin7x+N7cos7x)-10(Mcos7x+Nsin7x)=3cos7x$
Simplifying and the equating the coefficients relating sin and cos results in;
$-59M-21N=3$
$-59N+21M=0$
Solving for M and N results in;

  $M=-177/3922,N=-63/3922$

Problem Statement

Find the overall solution $y(x)$ that corresponds to the initial conditions $y(0)=1,y'(0)=0$. Plot over three periods.

Solution

From before, one period $=\pi /7$ so therefore, three periods is $3\pi /7.$
Using the roots given in the notes $\lambda _{1}=-2,\lambda _{2}=5$, the homogenous solution becomes;
$y_{h}(x)=c_{1}e^{-2x}+c_{2}e^{5x}$
Using initial condtion $y(0)=1$;
$1=c_{1}+c_{2}$
$y'_{h}(x)=-2c_{1}e^{-2x}+5c_{2}e^{5x}$
with $y'(0)=0$
$0=-2c_{1}+5c_{2}$
Solving for the constants;
$c_{1}=5/7,c_{2}=2/7$
$y_{h}(x)=5/7e^{-2x}+2/7e^{5x}$
Using the $y_{p}(x)$ found in the last part;
$y=y_{h}+y_{p}$

 $y=5/7e^{-2x}+2/7e^{5x}-177/3922cos7x-63/3922sin7x$