University of Florida/Egm6341/s10.team2.niki/HW7

problem 2: Solution of the Logistic equation

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Statement

P.39-1

Solution

We have Verhulst model or the Logistic equation as P.38-3

     ${\dot {x}}={\frac {dx}{dt}}=rx(1-{\frac {x}{x_{max}}})$

Separating variables and integrating we have

     $\int _{x_{0}}^{x}{\frac {x_{max}}{x(x_{max}-x)}}dx=\int _{t_{0}=0}^{t}rdt$

which can be written as

     $\int _{x_{0}}^{x}{\frac {1}{x}}dx+\int _{x_{0}}^{x}{\frac {1}{(x_{max}-x)}}dx=\int _{t_{0}=0}^{t}rdt$

We get

     $log_{e}({\frac {x}{x_{0}}})-log_{e}({\frac {x_{max}-x}{x_{max}-x_{0}}})=r(t-0)$

     ${\frac {x(x_{max}-x_{0})}{x_{0}(x_{max}-x)}}=e^{rt}$

Rearranging we have

     $x={\frac {x_{max}x_{0}e^{rt}}{x_{max}+x_{0}(e^{rt}-1)}}$

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problem 14: Constants of the Cosine Series

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Statement

We have the cosine series expressed as $f(cos\theta )={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }a_{k}cos(k\theta )$, we need to express teh constants $a_{k}$ as

$a_{k}={\frac {2}{\pi }}\int _{0}^{\pi }f(cos\theta )cos(k\theta )d\theta$

Solution

We have the given expression for the cosine series as

$f(cos\theta )={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }a_{k}cos(k\theta )$

multiplying both sides by $cos(m\theta )$ where k is not the same as m and integrating we get,

$\int _{0}^{2\pi }f(cos\theta )(cos(m\theta ))=\int _{0}^{2\pi }{\frac {a_{0}cos(m\theta )}{2}}+\int _{0}^{2\pi }\sum _{k=1}^{\infty }a_{k}cos(k\theta )cos(m\theta )$

Using the property of orthogonality we know that $\int _{0}^{2\pi }a_{k}cos(k\theta )cos(m\theta )$ exists only when k = m i.e

$\int _{0}^{2\pi }f(cos\theta )(cos(k\theta ))d\theta ={\frac {a_{0}}{2}}\int _{0}^{2\pi }cos(m\theta )d\theta +a_{k}\int _{0}^{2\pi }cos^{2}(k\theta )d\theta$

wkt,

$\int _{0}^{2\pi }cos(m\theta )d\theta =0$ and

$\int _{0}^{2\pi }cos^{2}(k\theta )d\theta ={\frac {2\pi -0}{2}}=\pi$

Thus we have by substituting and rearranging terms,

$a_{k}={\frac {2}{\pi }}\int _{0}^{\pi }f(cos\theta )cos(k\theta )d\theta$

Author

--Egm6341.s10.team2.niki 14:43, 23 April 2010 (UTC)

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