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Laplace Transform - Coggle Diagram
Laplace Transform
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4.2: Transformation of Initial Value Problems
NOTE: first on 2nd order linear with constant coefficients
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Definition (piecewise smooth function):
f is piecewise smooth on the bounded interval [a,b] if it is piecewise continuous on [a,b] and differentiable except at finitely many points, with f'(t) being piecewise continuous on [a,b].
Figure 4.2.4: Using the Laplace Transform to solve an initial value problem.
- Given a Diffy Q in terms of x(t)
- Use L.T. (and the initial conditions) to convert to algebratic eqn in X(s)
- Solve that algebraically
- Apply inverse L.T. to the solution X(s)
- That is the solution x(t) to the original Diffy Q
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4.4: Derivatives, Integrals, and Products of Transforms
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Theorem 1: The Convolution Property :question:
The Laplace transform of the convolution f(t) g(t) exists (assuming the conditions hold) therefore we can find the inverse transform of the product F(s) G(s).
This is an alternative to the partial fractions technique of finding the inverse transform
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