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Chapter 21: Problem 1

Express the following expressions as partial fractions, using complex numbers if necessary. Hence find their inverse Laplace transforms. (a) \(\frac{3 s-2}{s^{2}+6 s+13}\) (b) \(\frac{2 s+1}{s^{2}-2 s+2}\) (c) \(\frac{s^{2}}{\left(s^{2} / 2\right)-s+5}\) (d) \(\frac{s^{2}+s+1}{s^{2}-2 s+3}\) (e) \(\frac{2 s+3}{-s^{2}+2 s-5}\)

Short Answer

Expert verified
Each expression can be decomposed into partial fractions with complex roots, and the inverse Laplace transforms involve exponential functions with cosines and sines.

Step by step solution

01

Factor the Denominators

For each expression, we start by factoring the denominators. Some factors might include complex numbers. Denominators are given as quadratics:(a) \(s^2 + 6s + 13\)(b) \(s^2 - 2s + 2\)(c) \(s^2/2 - s + 5 = \frac{1}{2}(s^2 - 2s + 10)\)(d) \(s^2 - 2s + 3\)(e) \(-s^2 + 2s - 5\)
02

Find Roots of Quadratics

Find the roots of the quadratic equations obtained from the denominators to express the functions in partial fraction form. Use the quadratic formula \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find these roots.(a) Roots, \(-3 \pm 2i\)(b) Roots, \(1 \pm i\)(c) Roots, \(1 \pm 3i\)(d) Roots, \(1 \pm i\)(e) Roots, \(1 \pm 2i\)
03

Partial Fraction Decomposition

Express each function as a partial fraction using the roots found in step 2, corresponding to complex conjugate pairs. For each conjugate pair \((s - (a + bi))(s - (a - bi))\), use the form \( \frac{A}{s - (a+bi)} + \frac{B}{s - (a-bi)} \).(a) \(\frac{3s-2}{(s + 3 - 2i)(s + 3 + 2i)}\)(b) \(\frac{2s+1}{(s - 1 - i)(s - 1 + i)}\)(c) \(\frac{s^2}{(s - 1 - 3i)(s - 1 + 3i)}\)(d) \(\frac{s^2+s+1}{(s - 1 - i)(s - 1 + i)}\)(e) \(\frac{2s+3}{(-s + 1 - 2i)(-s + 1 + 2i)}\)
04

Solve for Partial Fraction Coefficients

Find the values of coefficients by comparing coefficients or using substitution. Equate the expressions by comparing both sides after expressing them using partial fractions.Example: If \(\frac{A}{s - (a + bi)} + \frac{B}{s - (a - bi)}\) equates to the original function, solve for \(A\) and \(B\) respecting real and imaginary parts.
05

Inverse Laplace Transform

Using the inverse Laplace transform for each partial fraction:1. Recognize that the inverse transform of \(\frac{1}{s-a}\) is \(e^{at}\).2. For complex roots, \(\frac{1}{s - (a+bi)}\), the transform involves \(e^{at}(cos(bt) + i\sin(bt))\).Compute the Inverse Laplace for each decomposed term from partial fractions found in earlier steps.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Fractions
The concept of partial fractions is essential in transforming complex algebraic expressions into a more manageable form. When you look at expressions like those in our examples, the goal is to break them down into simpler fractions which are easier to work with, especially when applying the Laplace Transform. Here's a quick guide to understand partial fractions:
  • Partial fraction decomposition involves expressing a rational function as a sum of simpler fractions.
  • Decomposition is particularly useful when the denominator consists of polynomial expressions that can be factored into simpler components.
  • For quadratics that do not factor neatly into real numbers, complex numbers may be necessary.
By converting a complex rational expression into partial fractions, you allow for simpler manipulation and evaluation of each part individually. This is especially advantageous in fields like control systems and differential equations.
Quadratic Roots
Quadratic roots play a pivotal role in breaking down expressions in the partial fraction process. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\) and its roots can be found using the quadratic formula:\[s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Understanding how to find these roots is crucial, especially when the discriminant \(b^2 - 4ac\) is negative, indicating complex roots. Here's how to deal with different scenarios:
  • Positive discriminant: The quadratic has two distinct real roots.
  • Zero discriminant: The quadratic has exactly one real root, known as a repeated root.
  • Negative discriminant: The roots are complex and occur in conjugate pairs.
These roots are the foundation for the partial fraction decomposition and will guide the formation of simpler expressions for inverse transforms.
Complex Numbers
Complex numbers are crucial when handling quadratics with negative discriminants. They are numbers of the form \( a + bi \), where \(i\) is the imaginary unit defined by \(i^2 = -1\). Complex numbers appear when the method of partial fraction decomposition involves solving quadratics with negative discriminants, resulting in imaginary numbers.Here鈥檚 why complex numbers matter:
  • Their introduction makes it possible to handle expressions that would otherwise not have real solutions.
  • Complex conjugate pairs, which are crucial in partial fraction decomposition, ensure real coefficients in solutions.
When working with Laplace Transforms, complex numbers extend your toolkit, allowing you to deal with a broader range of problems, ensuring that solutions are both meaningful and applicable in real-world scenarios like electrical engineering and signal processing.
Inverse Transform
Inverse Laplace Transform is the process of finding the original function from its Laplace Transform, often expressed in the frequency domain. This process is crucial when moving from analysis back to actual real-world signals or functions.Here's a step-by-step approach to understanding inverse Laplace transforms:
  • Each term in the decomposed partial fractions has an associated inverse transform.
  • The inverse transform of \(\frac{1}{s-a}\) is \(e^{at}\), a basic and fundamental form.
  • When dealing with complex roots \((a+bi)\), the inverse transform involves the exponential function combined with trigonometric functions: \(e^{at}(\cos(bt) + i\sin(bt))\).
The interplay between these transforms and how they bring back time-domain functions is essential in fields like signal processing, where understanding and interpreting data relies on transitioning smoothly between domains.
Complex Conjugates
Complex conjugates are pairs of complex numbers that are mirror images across the real axis. If you have a complex number \(a + bi\), its conjugate is \(a - bi\). In terms of partial fraction decomposition and Laplace Transforms, these conjugates ensure that expressions remain real and manageable.Here's why complex conjugates are important:
  • They guarantee that coefficients in the thousand answers to polynomial equations are real.
  • In partial fraction decomposition, conjugates help factorize quadratic expressions with complex roots effectively.
  • Handling conjugates allows a seamless transition between mathematical manipulations and the interpretation of physical phenomena.
Recognizing complex conjugates and using them efficiently is fundamental in ensuring those solutions remain grounded in real-world applications. It streamlines solving differential equations and is beneficial in engineering and physics contexts.

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Most popular questions from this chapter

Using Laplace transforms find the particular solution of $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}-5 \frac{\mathrm{d} y}{\mathrm{~d} t}-6 y=14 \mathrm{e}^{-t} $$ satisfying \(y=3\) and \(\frac{\mathrm{d} y}{\mathrm{~d} t}=8\) when \(t=0\).

Express the following as partial fractions and hence find their inverse Laplace transforms: (a) \(\frac{5 s+2}{(s+1)(s+2)}\) (b) \(\frac{3 s+4}{(s+2)(s+3)}\) (c) \(\frac{4 s+1}{(s+3)(s+4)}\) (d) \(\frac{6 s-5}{(s+5)(s+3)}\) (e) \(\frac{4 s+1}{s(s+2)(s+3)}\) (f) \(\frac{7 s+3}{s(s+3)(s+4)}\) (g) \(\frac{6 s+7}{s(s+2)(s+4)}\) (h) \(\frac{8 s-5}{(s+1)(s+2)(s+3)}\) (i) \(\frac{3 s+5}{(s+1)\left(s^{2}+3 s+2\right)}\) (j) \(\frac{2 s-8}{(s+2)\left(s^{2}+7 s+6\right)}\)

Find the Laplace transforms of (a) \(3 u(t)+\delta(t)\) (b) \(-6 u(t)+4 \delta(t)\) (c) \(3 u(t-2)+\delta(t-2)\) (d) \(u(t-3)-\delta(t-4)\) (e) \(\frac{1}{2} u(t-4)+3 \delta(t-4)\) where \(u(t)\) is the unit step function.

Find the Laplace transforms of the following functions: (a) \(3 t^{2}-4\) (b) \(2 \sin 4 t+11-t\) (c) \(2-t^{2}+2 t^{4}\) (d) \(3 \mathrm{e}^{2 t}+4 \sin t\) (e) \(\frac{1}{3} \sin 3 t-4 \cos \left(\frac{t}{2}\right)\) (f) \(3 t^{4} \mathrm{e}^{5 t}+t\) (g) \(\sinh 2 t+3 \cosh 2 t\) (h) \(\mathrm{e}^{-t} \sin 3 t+4 \mathrm{e}^{-t} \cos 3 t\)

Express the following fractions as partial fractions and hence find their inverse Laplace transforms: (a) \(\frac{3 s+3}{(s-1)(s+2)}\) (b) \(\frac{5 s}{(s+1)(2 s-3)}\) (c) \(\frac{2 s+5}{s+2}\) (d) \(\frac{s^{2}+4 s+4}{s^{3}+2 s^{2}+5 s}\) (e) \(\frac{1-s}{(s+1)\left(s^{2}+2 s+2\right)}\) (f) \(\frac{s+4}{s^{2}+4 s+4}\) (g) \(\frac{2\left(s^{3}-3 s^{2}+s-1\right)}{\left(s^{2}+4 s+1\right)\left(s^{2}+1\right)}\) (h) \(\frac{3 s^{2}-s+8}{\left(s^{2}-2 s+3\right)(s+2)}\)
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