Cy(t) (C is a constant)
n is a non-negative integer
|n!/sn+1||Done for n=0; done for n=1; done for n=2; statement for any n. Used very often afterwards.|
|eat||1/(s-a)||Verification. Used often.|
|Verification for cosine using complex exponents. The text proves this using integration by parts. Both results are used often. You need to know and believe in Euler's formula.|
|Not verified in class, but a discussion here indicates why sinh and cosh are useful.|
y(t)=t if t in [0,1] else y(t)=0.
|(-se-s-e-s+1)/s2||Computed from the definition. Students should also know how to do this with U and the translation theorems.|
|y(t) is a function|
with exponential growth
|Y(s)-->0 as s-->infinity.|
Y(s)-->the net area under all of the y(t) function as s-->0+.
|y´(t)||sY(s)-y(0)||Verified here. Used later frequently.|
|-4+sY(s)+2Y(s)=3s/(s2+1)||First use of Laplace transform method to solve an initial value problem. The steps (transform, solve, inverse transform) are used very often, so this should be considered a model. Here is a second order equation, and here is a link to a solution of an integral equation from the course I gave a year ago, since the choice I made this year didn't work out.|
|y(n)(t)||snY-sn-1y(0)-sn-2y´´(0)-...-y(n-1)(0)||First mentioned here and first used here to solve a second order ODE.|
|eatf(t)||F(s-a)||First translation theorem. Examples. Many other examples later, such as here and here.|
U(t)=0 if t<0 & =1 if t>=0.
|1/s||Defined here and its use in writing piecewise functions algebraically follows immediately afterwards.|
|f(t-a)U(t-a)||e-asF(s)||Second translation theorem. Examples.|
|g(t)U(t-a)||e-as multiplied by the Laplace transform of g(t+a).||Alternate form of the second translation theorem Examples. Many other examples later, such as here and here.|
|t·f(t)||F´(s)||Verification. Used here.|
|(-1)ntn·f(t)||(dn/dsn)F(s)||Just the statement|
|f*g(t)=0tf(tau)g(t-tau)dtau||F(s)·G(s)||Convolution defined. One and two direct computations of convolution. and here's the statement of the Laplace transform fact. And some simple uses. A more complicated use is attempted here.|
|delta(t-t0)||e-t0s||Verification here. Before and after this are substantial discussions of the "theory", and then some examples using delta in mathematical models.|
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