A basic introduction on the definition of the Laplace transform was given in this tutorial. While it is good to have an understanding of the Laplace transform definition, it is often times easier and more efficient to have a Laplace Transform table handy such as the one found here. One common Laplace transform that we will revisit in later examples, is the transform of a function's derivative. The Laplace transform of the 1st derivative of a function is defined as follows:
$$\mathcal{L}(f'(t)) = s\mathcal{L}(f(t)) - f(0) $$
More generally, the Laplace Transform of the nth derivative of a function is as follows:
$$\mathcal{L}(f^n(t)) = s^n\mathcal{L}(f(t)) - s^{n-1}f(0) - s^{n-2}f'(0)...- sf^{(n-2)}(0) - f^{(n-1)}(0) $$
The above two definitions are found in the table provided.
Laplace transforms can be used to solve initial value problems for linear differential equations with constant coefficients. Furthermore, real-world applications of the Laplace Transform are found in the analysis of mechanical vibrations and electrical circuits. The differential equations of such systems are shown below:
Equation of motion for a vibrating spring-mass system:
$$m\frac{d^2u}{dt^2} + \gamma \frac{du}{dt} + ku = F(t) $$
...where:
$$m = mass $$
$$\qquad \; \; \; \; \gamma = damping \; coefficient $$
$$ \qquad \; \; \; \; k = spring \; constant$$
$$ \qquad \; \; \; \; F(t) = Applied \; external \; force$$
Equation of an electrical circuit containing resistance (R), inductance (L) and capacitance (C):
$$L \frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{1}{C}Q = E(t) $$
...where:
$$\qquad \; \; \; Q(t) = Charge \; on \; the \; capacitor$$
$$\qquad \; \; \; \; E(t) = applied \; voltage $$
By taking the derivative of the above equation and noting that:
$$I(t) = current = \frac{dQ}{dt} $$
...we get:
$$L \frac{d^2I}{dt^2} + R\frac{dI}{dt} + \frac{1}{C}I = \frac{dE}{dt} $$
Note that for the circuit and spring-mass equations, initial conditions for u, Q or I must be given or determined.To see some applications of solving differential equations via Laplace Transforms, have a look at some of the examples in the RLC circuit analysis tutorials: Step Response of RLC circuits.
Next we will look at some examples of using Laplace transforms to solve differential equations.
Continue on to Differential Equations w/ Laplace transforms (example 1)