User:Egm4313.s12.team14.tonyhan/r5

= Problem 5 =

Given
$$\ W(f,g)=fg'-gf'$$

Find
1) Show that sin7x and cos7x are linearly independent via Wronskian and Gramian methods

2) Find 2 equations for the two unknowns M and N. Solve for M and N

3) Find the overall solution y(x) that corresponds to the initial condition. Plot over 3 periods

Solution
Wronskian Method


 * {| style="width:100%" border="0" align="left"

W(f,g)=fg'-gf' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Given
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle f(x)=cos(7x)$$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle g(x)=sin(7x)$$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle f'(x)=-7sin(7x)$$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle g'(x)=7cos(7x)$$
 * $$\displaystyle
 * $$\displaystyle
 * }

Dashpot Force:


 * {| style="width:100%" border="0" align="left"

\vec{F}_D=c(y')) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Applied Force:


 * {| style="width:100%" border="0" align="left"

\vec{F}(t) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Forces:


 * {| style="width:100%" border="0" align="left"

\Sigma\vec{F}=\vec{F}(t)-\vec{F}_s-\vec{F}_D $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }


 * {| style="width:100%" border="0" align="left"

m\vec{a}=\vec{F}_i=my'' $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }