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khurram usman

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Laplace can be used to analyze unstable systems.

Fourier is a subset of laplace.

Some signals have Fourier but laplace is not defined , for instance cosine or sine from -infinity to +infinity.

i have studied signals and systems and basic control theory in my undergrad. My question is related to the solution of differntial equations using these transforms.

Wherever i have seen it is written that Fourier is used for steady state analysis (example : in solution of circuits) whereas for transient response we resort to laplace. What exactly is the thing that enables laplace to incorporate initial conditions (hence unilateral laplace) and solve ODEs ? Similarly what prevents Fourier from taking these into account?

Let me explain at least what i understand. when solving for the laplace of the derivative of a function (using integration by parts) we input the initial conditions there and they usually end up appearing as decaying exponentials (at the natural modes/poles of the system) in the final response. The constants are chosen so as to satisfy the initial conditions. On a sidenote, this is also related to linearity and upon initial conditions that are not 0 (rest) the system ends up becoming non-linear because of not satisfying the zeros input zero output property. Now when the Fourier is found for the derivative of a function the limits of the integral go from -infinity to + infinity and consequently the initial conditions can't be absorbed. Why can't we define the integral to be from 0 to infinity and incorporate initial conditions just as in laplace? i know that Fourier is closely linked to convolution and changing these limits would wrong that but still can this formulation be used for solving an initial value ODE? if yes , kindly provide an example. here is one that i found . i don't understand one step (second step of finding the general solution'u') in it but other than that i can't find a mistake.

http://math.stackexchange.com/quest...ial-value-ode-problem-using-fourier-transform

i know this might be too basic for some or too narrowed down but it has bugged me my whole undergrad and now i really need an answer from an expert. Thank you and regards khurram