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It's a start. Example(s) will be completed later.
Laplace transforms are one of the ways of solving linear ordinary [[differential equation]]s (Linear ODEs) with [[constant]] coefficients. This technique allows us to transform a Linear ODE into a linear algebraic equation.
==Definition==
The unilateral Laplace transform is defined by
:<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math>
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==Example==
===Example 1===
Consider the following initial value problem
:<math>y'+y=e^{at}</math>
where <math>a</math> is constant, subject to
:<math>y(0)=0</math>
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>
Or
:<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
:<math>\int_0{\infty}e^{-st+at} \,dt=\lim{}</math>
-->
==References==
D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
{{stub}}
==Definition==
The unilateral Laplace transform is defined by
:<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math>
<!--
==Example==
===Example 1===
Consider the following initial value problem
:<math>y'+y=e^{at}</math>
where <math>a</math> is constant, subject to
:<math>y(0)=0</math>
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>
Or
:<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
:<math>\int_0{\infty}e^{-st+at} \,dt=\lim{}</math>
-->
==References==
D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
{{stub}}