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Calculus: Differential Equations (Part 5 of x)

Preface
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.

Introduction
This article will cover differential equations using Laplace Transforms.

There is no reference book for this entry.

This also assumes you are familiar with Python or can stumble your way through it.

Concepts
Dependent Variable: in a derivative the variable being derived (e.g. x in dx/dt).

Independent Variable: in a derivative the variable being derived against (e.g. t in dx/dt).

Coefficient: a number (or otherwise) multiplied by a variable.

Order: a number quantifying the largest degree of any derivative in an equation.

Ordinary Differential Equation: a differential equation only containing derivatives with respect to a single variable.

Partial Differential Equation: a differential equation only containing partial derivatives with respect to more than one variable.

Linear Differential Equation: any equation which 1) the dependent variables are of degree 1 or 0, 2) can contain any number of derivatives of the dependent variable, 3) is an additive combination and 4) the coefficients are not functions of the dependent variable.

Properties of Laplace Transforms

Linearity 1 L{g + h} = L{g} + L{h}
Linearity 2 L{cg} + cL{g}
Translation L{eatf(t)}(s)=F(s-a)
First Derivative L{f'}(s)=sL{f}(s)-f(0)
nth Derivative L{fn}(s)=snL{f}(s) - sn-1f(0) - sn-2f'(0) -...- fn-1(0)
Derivative of Laplace L{tnf(t)}(s)=(-1)n dnF / dsn (s)
Laplace of Unit Step Function L{f(t)u(t-a)}(s)=e-asL{g(t+a)}(s)
Laplace of Periodic Function L{fT}(s) = FT(s) / (1-e-sT)
Laplace of Dirac Delta Function L{δ(t-a)}(s) = e-as
Dirac Delta - Unit Step Relationship δ(t-a) = u'(t-a)

Where g and h are a functions, c is a constant.

Laplace Transform Table
Here.

Intuitive Understanding of Second Order Differential Equations
Many physical phenomena can be described as a second order differential equation.

General Form: my'' + by' + ky = Fexternal

Or more intuitively: [inertia] y'' + [damping] y' + [stiffness] y = [Fexternal],
where y is some physical measurement (e.g. displacement, travel, distance, etc).

The behavior of the system can be understood by analyzing the constants m, b, k and F.

The response of a system can be broken into the follow components:

  1. Transient Motion, or initial onset at t=0 and,
  2. Steady State, or long term behavior as t→∞

Steady State behavior is the behavior caused by the external forcing energy, F. It can take on any behavior (e.g. constant value, decaying or sinusoidal) and the system will often introduce phase shift offset and a gain multiplying factor.

The characteristics of the transient response are a function of the system itself and fall into the following categories:

  1. Underdamped or Oscillitory Motion (b2 < 4mk)
  2. Overdamped Motion (b2 > 4mk)
  3. Critically Damped Motion (b2 = 4mk)

There are two very important points:

  • It is interesting to note that critically damped motion reaches steady faster than either underdamped or underdamped behavior.
  • The gain of a system is a function of frequency. The frequency where maximum gain occurs is called resonance.

Existence and Uniqueness Theorem
For a differential equation of order n and

  1. p1(x),...,pn(x) are all continuous on an interval (a,b) that contains the point x0
  2. g(x) is continuous on an interval (a,b) that contains the point x0
  3. initial values λ0,...,λn-1

then there exists a unique solution y(x) on the whole interval (a,b) to the problem:

  1. yn(x) + p1y(n-1)(x) + ... + pny(x) = g(x)
  2. y(x0) = λ0, ... , y(n-1)(x0) = λn-1

of the form

y(x) = yh + yp = C1y1(x) + ... + Cnyn(x) + yp(x)

Standard Solution to Linear Differential Equations
Let

  1. [y1, ... , yn] be the linearly independent solution set on (a,b) of yn(x) + p1y(n-1)(x) + ... + pny(x) = 0
  2. yp(x) be the particular solution to yn(x) + p1y(n-1)(x) + ... + pny(x) = g(x)
  3. p1, ... , pn are continuous on (a,b)

then the solution to the differential equation is
y(x) = yp(x) + C1y1(x) + ... + Cnyn(x).

Laplace Transform Definition
Let f(t) be a function on [0,∞). The Laplace transform of f is the function F defined by the integral:

F(s) = ∫0 e-stf(t)dt.

The domain of F(s) is all the values of s for which the integral exists.

The notation for a Laplace transform is given by L{f}.

Conversely, for function f(t), continuous on [0,∞), the inverse of F(s) is that which satisfies:

L{f} = F,

The notation for the inverse Laplace is given by L-1{F}.

Solving Initial Value Problems with Laplace Transforms

  1. Derive the equation from the problem.
  2. Take the Laplace transform of both sides of the equation.
  3. Plug in the initial values. If this is a differential equation you will have fn(0) constants.
  4. Take the inverse Laplace transform.

Convolution Definition
Let f(t) and g(t) be continuous on [0,∞). The convolution is defined by:

(f*g)(t) = ∞0t f(t-v)g(v)dv = F(s)G(s)

Transfer Function
The relationship of the output of a system to its input can be defined by:

H(s)=Y(s)/G(s)

The impulse response, or natural response of a system defined by H(s) is given by:

h(t)=L-1{H}(t).

Next Up
This series is completed.

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