y_{1}(t)+y_{2}(t) Cy(t) (C is a constant) |
Y_{1}(s)+Y_{2}(s) CY(s) | |
t^{n} n is a non-negative integer | n!/s^{n+1} | Done for n=0; done for n=1; done for n=2; statement for any n. Used very often afterwards. |
e^{at} | 1/(s-a) | Verification. Used often. |
sin(kt) cos(kt) | k/(s^{2}+k^{2}) s/(s^{2}+k^{2}) | 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. |
sinh(kt) cosh(kt) | k/(s^{2}-k^{2}) s/(s^{2}-k^{2}) | 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)/s^{2} | 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. |
Solving
1y´(t)+2y(t)=3cos(t) with y(0)=4 | -4+sY(s)+2Y(s)=3s/(s^{2}+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) | s^{n}Y-s^{n-1}y(0)-s^{n-2}y´´(0)-...-y^{(n-1)}(0) | First mentioned here and first used here to solve a second order ODE. |
e^{at}f(t) | F(s-a) | First translation theorem. Examples. Many other examples later, such as here and here. |
U(t) 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^{-as}F(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)^{n}t^{n}·f(t) | (d^{n}/ds^{n})F(s) | Just the statement |
f_{*}g(t)=_{0}^{t}f(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. |
_{0}^{t}f(tau)dtau | F(s)/s | Verified here |
delta(t-t_{0}) | e^{-t0s} | Verification here. Before and after this are substantial discussions of the "theory", and then some examples using delta in mathematical models. |
Maintained by greenfie@math.rutgers.edu and last modified 9/20/2005.