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

$y_p(x)=Mcos7x+Nsin7x$
$y'_p(x)=-M7sin7x+N7cos7x$
$y''_p(x)=-M7^2cos7x-N7^2sin7x$
Plugging these values into the equation given ($y''-3y'-10y=3cos7x$) yields;
$-M7^2cos7x-N7^2sin7x-3(-M7sin7x+N7cos7x)-10(Mcos7x+Nsin7x)=3cos7x$
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;
$y_h(x)=c_1e^{-2x}+c_2e^{5x}$
Using initial condtion $y(0)=1$;
$1=c_1+c_2$
$y'_h(x)=-2c_1e^{-2x}+5c_2e^{5x}$
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;
$y=y_h+y_p$

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