ReviewSheet2_S08 - • The Laplace transform 1 The...

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Review Sheet for Exam 2, Math 220 What we’ve covered since the last exam: The method of undetermined coefficients for linear differential equations with constant coefficients. The only tricky thing is recognizing the special case where the right hand side contains a solution to the homogeneous equation. Simple mechanical models - spring - mass systems with friction. Solving the IVP, recognizing the steady-state and transient solutions; resonance ; amplitude modulation. You should be able to write the steady state solution in the form A cos( ωt - φ ). Linear (LRC) electric circuits: Kirchoff’s laws for the voltages and currents. Writing down and solving the ODEs relating pairs of these quantities. Higher order linear ODEs with constant coefficients.
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Unformatted text preview: • The Laplace transform 1. The definition: L [ f ] = Z ∞ f ( t ) e-st dt. When in doubt, you can always compute something using this definition. 2. The inverse transform: L-1 ( Y ) = y ⇐⇒ L ( y ) = Y . 3. Solving an ODE with constant coefficients using Laplace transforms requires – Knowing how to transform derivatives – Knowing the two shift theorems: (1) L ( e-at f ( t )) = F ( s + a ), and (2) L ( u a ( t ) f ( t-a )) = e-as F ( s ) . – Using partial fractions. – Knowing how to transform step functions. 4. The unit impulse function and the transfer function. (If we have class on Tuesday) 1...
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