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expanding example; will be completed later
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==Example==
==Examples==
===Example 1===
===Example 1===
Consider the following initial value problem
Consider the following initial value problem
To solve this problem using laplace transform, first apply laplace transform to both sides of the equation, obtaining:
To solve this problem using laplace transform, first apply laplace transform to both sides of the equation, obtaining:
:<math>\int_0{\infty}e^{-st} (y'+y) \,dt= \int_0{\infty}e^{-st} \e^{at} \,dt</math>
:<math>\int_0^{\infty}e^{-st} (y'+y) \,dt= \int_0^{\infty}e^{-st} e^{at} \,dt</math>
Or
Or
:<math>\int_0{\infty}e^{-st}y'\,dt + \int_0{\infty}e^{-st}y'\,dt = \int_0{\infty}e^{-st+at} \,dt</math>
:<math>\int_0^{\infty}e^{-st}y'\,dt + \int_0^{\infty}e^{-st}y\,dt = \int_0^{\infty}e^{-st+at} \,dt</math>
The integral on the right hand side is
The integral on the right hand side is
:<math>\int_0{\infty}e^{-st+at} \,dt=\lim{}</math>
:<math>\int_0^{\infty}e^{-st+at} \,dt</math>
:<math>=\lim_{b\to\infty} \int_0^b e^{-st+at} \,dt</math>
:<math>=\lim_{b\to\infty}\left[\frac{1}{a-s} e^{(a-s)t} - \frac{1}{a-s}e^{(a-s)(0)}\right]</math>
:<math>= \lim_{b\to\infty}\left[\frac{1}{a-s} e^{(a-s)t} - \frac{1}{a-s}\right]</math>
If <math>s>a</math>,
:<math>\lim_{b\to\infty} \left[\frac{1}{a-s}e^{(a-s)t} - \frac{1}{a-s}\right]= \frac{1}{a-s}</math>
For the left side, if we apply integration by parts,
:<math>\int uv'\,dt=uv-\int u'v\,dt</math>
Substitution into the left side will get
:<math>\int_0^{\infty}e^{-st}y'\,dt + \int_0^{\infty}e^{-st}y'\,dt </math>
:<math>=e^{-st}y + s\int_0^{\infty}e^{-st}y\,dt + \int_0^{\infty}e^{-st}y\,dt </math>
:<math>=e^{-st}y + (s+1) \int_0^{\infty}e^{-st}y\,dt </math>
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==References==
==References==
D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*[http://mathworld.wolfram.com/LaplaceTransform.html Laplace transform] on Wolfram Mathworld
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