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

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. $$ = \int_{a}^{b}f(x)g(x)dx$$ $$\Gamma(f,g)=det\begin{bmatrix}  & \\  &  \end{bmatrix}$$ $$\int_{0}^{\pi/7}cos^2(7x)dx=\pi/14$$ $$\int_{0}^{\pi/7}sin^2(7x)dx=\pi/14$$ $$\int_{0}^{\pi/7}cos(7x)*sin(7x)dx=0$$ $$\Gamma(f,g)=det\begin{bmatrix} \pi/14 & 0\\ 0 & \pi/14 \end{bmatrix}$$ $$\Gamma(f,g)=\pi^2/49$$ 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; $$y=y_h+y_p$$ $$y=5/7e^{-2x}+2/7e^{5x}-177/3922cos7x-63/3922sin7x$$