Laplace Transforms

Exercise 1 : Applying Laplace Transforms

Obtain the Laplace transforms for the following functions:

  1. $\frac{1}{4}t^3$
  2. $5e^{-2t}$
  3. $4\cos 3t$
  4. $\sin^2 t$

Exercise 2 : Inverse Laplace Transforms

Obtain the inverse transforms for the following:

  1. $\frac{1}{4s^2 + 1}$
  2. $\frac{1}{s(s + 4)}$
  3. $\frac{2}{s(s^2 + 9)}$
  4. $\frac{6}{1 - s^2}$
  5. $\frac{1}{s^2(s^2 + 1)}$
  6. $\frac{4}{s(s^2 - 6s + 8)}$

Exercise 3 : Solving Linear ODEs via Laplace

Solve the following linear differential equations with constant coefficients:

  1. $\ddot{y} + 5\dot{y} + 4y = 0$ (initial conditions: $y = 0$, $\dot{y} = 2$ at $t = 0$)
  2. $\ddot{y} + 9y = \sin 2t$ (initial conditions: $y = 1$, $\dot{y} = -1$ at $t = 0$)
  3. $\ddot{y} + 2\dot{y} = \cos t$ (initial conditions: $y = 1$ at $t = 0$)

Exercise 4 : Transform and Solve (Verification)

If $\ddot{y} - 3\dot{y} + 2y = 4$ and $y = 2$, $\dot{y} = 3$ at $t = 0$, show that $\bar{y} = \frac{2s^2 - 3s + 4}{s(s - 1)(s - 2)}$, and hence find the solution for $y$.

Exercise 5 : Nonhomogeneous Forcing (Exponentials and Polynomials)

Given $\ddot{y} + \dot{y} = e^t + t + 1$ with the initial conditions $y = 0$, $\dot{y} = 0$ at $t = 0$, obtain $y$.

Exercise 6 : Coupled First-Order System (Forced)

Solve the following simultaneous equations for $y$:

$\dot{y} + 2\dot{x} + y - x = 25e^t$ $2\dot{y} + x = 25e^t$ Initial conditions: $y = 0$, $x = 25$ at $t = 0$.

Exercise 7 : Coupled Homogeneous System

Solve for $y$ and $x$ given:

$4\dot{x} - \dot{y} + x = 1$ $4\dot{x} - 4\dot{y} - y = 0$ Initial conditions: $x = 0$, $y = 0$ at $t = 0$.

Exercise 8 : Series LC Circuit (Forced Sinusoid)

An electrical circuit consists of a capacitor, $C$ farads, and an inductor, $L$ henries, in series, to which a voltage $E \sin \omega t$ is applied. If $Q$ is the charge on the capacitor in coulombs, show that

$$ Q = \frac{E}{L(\omega^2 - 1/LC)} \left[ \frac{\omega}{s^2 + 1/LC} - \frac{\omega}{s^2 + \omega^2} \right], \quad \omega^2 LC \neq 1 $$

and hence calculate $Q$ given that $C = 50 \times 10^{-6}F$, $L = 0.1H$, $\omega = 500 \frac{rad}{s}$, $E = 2V$, and $Q = \dot{Q} = 0$ at $t = 0$.

Exercise 9 : Unit Step and Time-Shifting (Heaviside)

  1. Find the Laplace transforms of $u(t-a)$ and $u(t-a)f(t-a)$ where $u$ is the unit step and $a>0$.
  2. Compute the inverse transform of $\frac{e^{-2s}}{s(s+1)}$ and write the piecewise time-domain expression.
  3. Solve $\ddot{y}+y = u(t-\pi)$ with zero initial conditions using Laplace transforms.

Exercise 10 : Dirac Delta and Impulse Response

  1. Compute $\mathcal{L}{\delta(t-a)}$ and $\mathcal{L}{\delta’(t)}$.
  2. Solve $\ddot{y}+3\dot{y}+2y=\delta(t-1)$ with zero initial conditions; interpret the solution as an impulse response.
  3. If a system has impulse response $h(t)=e^{-t}u(t)$, find its response to input $x(t)=\delta(t-2)$.

Exercise 11 : Convolution Theorem and System Response

  1. Use convolution to find the inverse Laplace transform of $\frac{1}{(s+1)(s^2+1)}$.
  2. Given $H(s)=\frac{1}{s^2+2s+2}$ and input $x(t)=\sin t,u(t)$, compute the output $y(t)=h*x$ using Laplace-domain multiplication and inverse transform.

Exercise 12 : Initial and Final Value Theorems

  1. Use the initial and final value theorems to evaluate $\lim_{t\to0^+}f(t)$ and $\lim_{t\to\infty}f(t)$ for $F(s)=\frac{s+2}{s^2+3s+2}$, if the limits exist.
  2. Apply the theorems to determine the steady-state value of the solution of $\dot{y}+2y=5$ with $y(0)=0$ via Laplace transforms.

Exercise 13 : Transfer Functions, Poles and Stability

  1. For the ODE $\ddot{y}+4\dot{y}+5y = u(t)$, find the transfer function $H(s)=Y(s)/U(s)$ and list its poles. Is the system stable?
  2. Find the natural response and comment on damping.

Exercise 14 : Periodic Forcing and Steady-State Response

  1. A square-wave of period $T$ with amplitude 1 can be expressed as a Fourier series. Briefly outline how Laplace transforms and the series combine to find the steady-state response of a linear system.
  2. As a concrete task, find the Laplace transform of a periodic pulse train of period $2$ consisting of unit pulses of width $0.5$.

Exercise 15 : Advanced Partial Fractions Practice

  1. Compute inverse Laplace transforms for the following using partial fractions:
    • $\frac{3s^2+2s+1}{s(s^2+2s+2)}$
    • $\frac{s^3+2s^2+s+1}{s^2(s+1)(s^2+1)}$
  2. Solve $\dddot{y}+\ddot{y}+\dot{y}+y=\sin t$ with zero initial conditions using transforms and partial fractions.