Laplace Transforms: Theory, Properties, and Applications to Differential Equations

Vocabulary practice flashcards defining core terms, fundamental theorems, operational identities, and solution procedures for Laplace transforms and differential equations.

Laplace Transform

An integral transform defined for a function $f(t)$ ($t \ge 0$) by the improper integral $F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t)\,dt$, which converts a function from the time domain $t$ into a function $F(s)$ in the frequency domain $s$.

Kernel of an Integral Transform

A function of two variables $k(s, t)$ that characterizes an integral transform $F(s) = \int_0^\infty k(s, t) f(t)\,dt$; for the Laplace transform, this kernel is $k(s, t) = e^{-st}$.

Inverse Laplace Transform

The operation denoted by $\mathcal{L}^{-1}$ that recovers the original time-domain function $f(t)$ from its frequency-domain transform $F(s)$, written as $f(t) = \mathcal{L}^{-1}\{F(s)\}$, satisfying $\mathcal{L}^{-1}\{\mathcal{L}\{f\}\} = f$ and $\mathcal{L}\{\mathcal{L}^{-1}\{F\}\} = F$.

Linearity of the Laplace Transform

The property stating that for any functions $f(t)$ and $g(t)$ whose transforms exist and any constants $a$ and $b$, the transform of their linear combination satisfies \mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)}.

First Shifting Theorem ($s$-Shifting)

The theorem stating that if $\mathcal{L}\{f(t)\} = F(s)$ for $s > k$, then multiplying the original function by an exponential factor $e^{at}$ shifts the transform parameter: $\mathcal{L}\{e^{at} f(t)\} = F(s - a)$ for $s - a > k$, with the inverse form $e^{at} f(t) = \mathcal{L}^{-1}\{F(s - a)\}$.

Existence Theorem for Laplace Transforms

The theorem stating that if $f(t)$ is defined and piecewise continuous on every finite interval of $t \ge 0$ and satisfies an exponential growth restriction for all $t \ge 0$ with constants $M$ and $k$, then the Laplace transform $\mathcal{L}(f)$ exists for all $s > k$.

Laplace Transform of the First Derivative

The operational formula stating that if $f(t)$ is continuous for all $t \ge 0$, satisfies the exponential growth restriction, and $f'(t)$ is piecewise continuous on every finite interval of $t \ge 0$, then $\mathcal{L}\{f'\} = s \mathcal{L}\{f\} - f(0)$.

Laplace Transform of the Second Derivative

The operational formula stating that if $f(t)$ and $f'(t)$ are continuous for all $t \ge 0$, satisfy the exponential growth restriction, and $f''(t)$ is piecewise continuous on every finite interval of $t \ge 0$, then $\mathcal{L}\{f''\} = s^2 \mathcal{L}\{f\} - s f(0) - f'(0)$.

Laplace Transform of the Derivative of Arbitrary Order

The general formula stating that if $f, f', \dots, f^{(n-1)}$ are continuous for all $t \ge 0$ and satisfy the growth condition, while $f^{(n)}$ is piecewise continuous on every finite interval of $t \ge 0$, then $\mathcal{L}\{f^{(n)}\} = s^n \mathcal{L}\{f\} - s^{n-1} f(0) - s^{n-2} f'(0) - \dots - f^{(n-1)}(0)$.

Laplace Transform of an Integral

The property stating that integration in the time domain corresponds to division by $s$ in the frequency domain: $\mathcal{L}\left\{\int_0^t f(\tau)\,d\tau\right\} = \frac{1}{s} F(s)$, or inversely $\int_0^t f(\tau)\,d\tau = \mathcal{L}^{-1}\left\{\frac{1}{s} F(s)\right\}$, valid for $s > 0$, $s > k$, and $t > 0$.

Subsidiary Equation

The algebraic equation in $Y(s) = \mathcal{L}\{y(t)\}$ obtained by taking the Laplace transform of a differential equation with given initial conditions, typically having the form $(s^2 + as + b)Y(s) = (s + a)y(0) + y'(0) + R(s)$.

Transfer Function ($Q(s)$)

The rational algebraic function defined by $Q(s) = \frac{1}{s^2 + as + b}$, which represents the ratio of the Laplace transform of the output to that of the input under zero initial conditions: $Q(s) = \frac{Y(s)}{R(s)} = \frac{\mathcal{L}(\text{output})}{\mathcal{L}(\text{input})}$.

Unit Step Function (Heaviside Function)

The function denoted $u(t - a)$ that equals $0$ when $t < a$ and $1$ when $t > a$ ($a \ge 0$), with a step jump of size $1$ at $t = a$, having the transform $\mathcal{L}\{u(t - a)\} = \frac{e^{-as}}{s}$ for $s > 0$.

Second Shifting Theorem ($t$-Shifting)

The theorem stating that if $\mathcal{L}\{f(t)\} = F(s)$, then a time delay of $a$ units ($a > 0$) corresponds to multiplication by $e^{-as}$: $\mathcal{L}\{f(t - a) u(t - a)\} = e^{-as} F(s)$, or inversely $f(t - a) u(t - a) = \mathcal{L}^{-1}\{e^{-as} F(s)\}$.

Alternative Second Shifting Formula

The formula used to transform an unshifted function multiplied by a step function: \mathcal{L}\{f(t) u(t - a)\} = e^{-as} \mathcal{L}\{f(t + a)}.

Gamma Function

The definite integral function defined by $\Gamma(a + 1) = \int_0^\infty e^{-x} x^a\,dx$, which gives the Laplace transform of fractional power functions $\mathcal{L}\{t^a\} = \frac{\Gamma(a + 1)}{s^{a + 1}}$ ($a > 0$) and satisfies $\Gamma(n + 1) = n!$ for non-negative integers $n$.

Three Steps of the Laplace Transform Method for ODEs

The three-step procedure: Step 1 sets up the subsidiary algebraic equation by transforming the ODE and applying initial conditions; Step 2 solves the subsidiary equation algebraically for $Y(s)$; Step 3 inverts $Y(s)$ to find the solution $y(t) = \mathcal{L}^{-1}\{Y(s)\}$.