Differential Equations: Laplace Transforms

Compute the inverse Laplace transform of the expression $\frac{2}{(s+1)^3}$.

Find the inverse Laplace transform of $\frac{s}{s^2 - 2s + 5}$ using the method of completing the square.

Compute the inverse Laplace transform of a rational function using partial fractions when the denominator has repeated linear factors.

Find the inverse Laplace transform of $\frac{3s + 8}{s^2 + 2s + 5}$

Find the inverse Laplace transform of the function $f(s) = \frac{1}{s - 3} - \frac{16}{s^2 + 9}$.

Solve the differential equation using Laplace transforms: $y'' + 5y' + 6y = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$.

Consider the differential equation $y'' - 2y' + 5y = 0$ with initial conditions $y(0) = 2$ and $y'(0) = -4$. Solve the initial value problem using the Laplace transform.

Solve the differential equation $y' + 4y = e^{4t}$ using Laplace transforms, given the initial condition $y(0) = 3$.

Solve a second order non-homogeneous ordinary differential equation (ODE) using the Laplace Transform with given initial conditions.

Find the Laplace transform of a piecewise function using unit step functions.

Solve the differential equation using the Laplace transform, considering the unit step function with initial conditions provided.

Given a piecewise function $f(t) = t$ for $0 \, \leq \, t \, < \, a$ and $0$ otherwise, express this function using step functions and compute its Laplace Transform.

Using a Laplace Transform, solve the second-order linear differential equation of motion for a damped harmonic oscillator and rearrange it for a transfer function.