Laplace Transform and Inverse Laplace Transform

Apply the Laplace transform definition to a constant function via an improper integral and limits. Conclude that the transform equals c over s for any constant c.

Explore the Laplace transform of monomial functions, from constants to higher powers, using integration by parts and limits. Derive L{1}=1/s and L{t}=1/s^2, and extend to higher monomials.

Apply the Laplace transform to exponential functions using the definition from 0 to infinity, yielding 1/(s−a) for e^{a t} and addressing the negative exponent case.

Explore the Laplace transform of trig functions by deriving the transforms of sine and cosine through integration by parts and s-domain expressions.

We derive the Laplace transform of hyperbolic functions by applying integration by parts from 0 to infinity, yielding a formula involving s^2 and a^2.

Compute the Laplace transform of a piecewise continuous function by splitting the integral from 0 to 1 and from 1 to infinity, applying the Laplace transform definition and evaluating.

This lecture derives the Laplace transform of a piecewise function with f(t)=0 on [0,1) and f(t)=t for t≥1, yielding F(s)=e^{-s}(1/s+1/s^2).

Derive the Laplace transform of a monomial by an iterative process, using the definition and integration by parts to prove L{t^n} = n! / s^{n+1} for n ≥ 0.

Explore the Laplace transform of monomial functions using the direct method, applying the formula directly, leveraging linearity to handle polynomials and integer powers, and noting gamma function for rational powers.

Apply the direct method to compute Laplace transforms of exponential functions, using the formula 1/(s−a) (or 1/(s+a)) and the linearity of transforms to simplify examples.

Explore direct method to transform sine and cosine using L{sin at}=a/(s^2+a^2) and L{cos at}=s/(s^2+a^2), with examples like sin 3t and sin(1/2)t.

Explore the direct method for Laplace transforms of hyperbolic functions. Use square minus a square formulas on sinh and cosh with examples like sinh(4x) and sinh(3x/2) to obtain simple results.

Express sinh x as (e^x - e^{-x})/2 and apply the linear Laplace transform to get 1/(s^2 - 1). Use the exponential form to verify the result.

Learn to use sum-to-product identities for sine and cosine with parameters alpha and beta to compute the Laplace transform of trigonometric product terms.

Explore the first shifting theorem for Laplace transforms, applying the exponential shift property to transform e^{at} f(t) into F(s−a) and verify with examples.

Apply the first shifting theorem of the Laplace transform by replacing s with s minus a for functions multiplied by e^{a} t, illustrated with exponentials, sines, cosines, and hyperbolic forms.

Explore the unit step function, defined as zero for t < a and one for t ≥ a, and learn its shifting properties for time-delayed signals in electrical engineering.

Proving the second shifting theorem for Laplace transforms, this lecture shows how a time shift with a step function yields a factor e^{-a s} in the s-domain, via substitution.

Demonstrates applying the second translation theorem to compute Laplace transforms of shifted and step-function inputs, including exponential factors and periodic signals.

Demonstrate the derivative property of the Laplace transform by differentiating the transform with respect to s, using the definition and the zero to infinity integral.

Explore how to differentiate Laplace transforms with respect to s through three worked examples, using exponentials and trigonometric forms to show how higher-order derivatives relate to time-domain functions.

Learn how the Laplace transform and inverse Laplace transform apply to simple constants and general functions, using standard formulas, shifting, and worked examples.

Explore the Laplace transform and its inverse by deriving and applying the standard exponential forms, including 1/(s−a) and 1/(s+a), with examples like 1/(s+3) and transforms of exponentials.

learn the laplace transform and its inverse, with the example 1/(s^2 + a^2) and the resulting forms for related functions.

Learn how to apply the inverse Laplace transform using hyperbolic functions, determining when the sign between terms and constants in the numerator dictate which hyperbolic form to use, with examples.

Explore three inverse Laplace transform examples, applying Laplace and inverse formulas to solve each function.

Learn how to compute inverse Laplace transforms via type 1 partial fractions for proper rational functions with linear factors, solving for A and B to get -e^{t}+e^{2t}.

Use partial fraction type 2 for a repeated linear factor such as (s-3)^2, solve for coefficients, and apply standard inverse Laplace transforms.

This lecture shows inverse Laplace transforms via partial fractions of type three when a linear factor coexists with an irreducible quadratic, solving for a, b, c and applying inverse transforms.

Apply the Laplace transform to derivatives with a proof, using the definition and integration by parts, to derive formulas for first and second derivatives and solve differential equations.

Learn to solve a linear first-order differential equation using the Laplace transform and its inverse. Apply initial conditions and convert derivatives to algebraic terms to obtain the solution.

Apply the Laplace transform to solve a first-order linear differential equation, using the initial condition y(0)=2 and inverse transform to obtain y(t).