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Calculus: Differential Equations (Part 5 of 5)

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 Definition
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.

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} = F.

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

Fourier Transform Definition
tbd

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)
Convolution 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)

Where g and h are a functions, c and a are constants.

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.

In the time domain, 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).

Solving Initial Value Problems with Laplace Transforms

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

Linear Systems of Differential Equations in Normal Form

x'(t) = A(t)x(t) + g(t),

where x(t) is the set of variables, g(t) is the forcing function and A(t) is an n x n matrix.

For equations of >1 order variable substitution is used (i.e. x(t)).

All the precepts from linear algebra apply here wrt existence and uniqueness. If those requirements are satisfied and the system is in normal form then the methods below can be used to solve the system.

Solving systems of Linear Differential Equations
General approach to solving systems of linear differential equations:

  1. Rewrite the system in general form (i.e. x'(t)=A(t)x(t) + g(t))
  2. Solve for the homogeneous solution for A(t)x(t)
  3. Solve for the non-homogeneous solution g(t)
  4. The general solution is a linear combination of all the non-homogeneous and homogeneous coefficients (i.e. x(t) = xp + xh)
  5. Apply the initial conditions for the explicit solution

To find the homogeneous solution:

  1. Find the eigenvalues of the system
  2. Find the eigenvectors of the system
  3. Substitute the eigenvalues and the eigenvectors to form the general solution of:
    x=C1k1er1t + C2k2er2t + ... + Cnknernt
  4. To determine the specific solution apply the initial values to the general solution.
  5. Solve the resulting system and apply those values to the general solution.

To find the homogeneous solution:

  1. There are some methods available, none I like.

Alternatively you can apply Laplace transformations to the system to obtain a solution:

  1. Write out the system explicitly
  2. Take the Laplace of all equations.
  3. Put the equations in the form of Ax=b.
  4. Find the inverse of A.
  5. Solve for x(s).
  6. Take the inverse of x(s) to solve for x(t).

This actually works quite well to solve for xh and xp simultaneously. The challenge is working symbolically and obtaining both inverses and that may be prohibitive.

Partial Differential Equations
Partial differential equations are simply functions involving several variables and their partial derivatives. In practice they are much more difficult to solve.

All the principles of linearity, existence and uniqueness apply to PDEs.

Next Up
This series is concluded.

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