ODEs: Classification Of Differential Equations

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• Front Matter
  • Colophon
  • Abstract
• 0 Introduction
  • 0.1 About this book
    • 0.1.1 Math 204: Vector Calculus and Differential Equations
    • 0.1.2 Original Textbook by Jiří Lebl
    • 0.1.3 Differences between the versions
    • 0.1.4 The video playlist
    • 0.1.5 Computer resources
    • 0.1.6 Acknowledgments
  • 0.2 What are Differential Equations?
    • 0.2.1 Differential equations
    • 0.2.2 Solutions of differential equations
    • 0.2.3 Differential equations in practice
    • 0.2.4 Four fundamental equations
    • 0.2.5 Exercises
  • 0.3 Classification of differential equations
    • 0.3.1 Exercises
• 1 First order equations
  • 1.1 Integrals as solutions
    • 1.1.1 Exercises
  • 1.2 Slope fields
    • 1.2.1 Slope fields
    • 1.2.2 Existence and uniqueness
    • 1.2.3 Exercises
  • 1.3 Separable equations
    • 1.3.1 Separable equations
    • 1.3.2 Implicit solutions
    • 1.3.3 Examples of separable equations
    • 1.3.4 Exercises
  • 1.4 Linear equations and the integrating factor
    • 1.4.1 Exercises
  • 1.5 Substitution
    • 1.5.1 Substitution
    • 1.5.2 Bernoulli equations
    • 1.5.3 Homogeneous equations
    • 1.5.4 Exercises
  • 1.6 Autonomous equations
    • 1.6.1 Exercises
  • 1.7 Numerical methods: Euler’s method
    • 1.7.1 Exercises
  • 1.8 Exact equations
    • 1.8.1 Solving exact equations
    • 1.8.2 Integrating factors
    • 1.8.3 Exercises
  • 1.9 First order linear PDE
    • 1.9.1 Exercises
• 2 Higher order linear ODEs
  • 2.1 Second order linear ODEs
    • 2.1.1 Exercises
  • 2.2 Constant coefficient second order linear ODEs
    • 2.2.1 Solving constant coefficient equations
    • 2.2.2 Complex numbers and Euler’s formula
    • 2.2.3 Complex roots
    • 2.2.4 Exercises
  • 2.3 Higher order linear ODEs
    • 2.3.1 Linear independence
    • 2.3.2 Theory of Higher Order ODEs
    • 2.3.3 Constant coefficient higher order ODEs
    • 2.3.4 Exercises
  • 2.4 Mechanical vibrations
    • 2.4.1 Some examples
    • 2.4.2 Free undamped motion
    • 2.4.3 Free damped motion
      • 2.4.3.1 Overdamping
      • 2.4.3.2 Critical damping
      • 2.4.3.3 Underdamping
    • 2.4.4 Exercises
  • 2.5 Nonhomogeneous equations
    • 2.5.1 Solving nonhomogeneous equations
    • 2.5.2 Undetermined coefficients
    • 2.5.3 Variation of parameters
    • 2.5.4 Exercises
  • 2.6 Forced oscillations and resonance
    • 2.6.1 Undamped forced motion and resonance
    • 2.6.2 Damped forced motion and practical resonance
    • 2.6.3 Exercises
• 3 The Laplace transform
  • 3.1 The Laplace transform
    • 3.1.1 The transform
    • 3.1.2 Existence and uniqueness
    • 3.1.3 The inverse transform
    • 3.1.4 Exercises
  • 3.2 Transforms of derivatives and ODEs
    • 3.2.1 Transforms of derivatives
    • 3.2.2 Solving ODEs with the Laplace transform
    • 3.2.3 Using the Heaviside function
    • 3.2.4 Transfer functions
    • 3.2.5 Transforms of integrals
    • 3.2.6 Exercises
  • 3.3 Convolution
    • 3.3.1 The convolution
    • 3.3.2 Solving ODEs
    • 3.3.3 Volterra integral equation
    • 3.3.4 Exercises
  • 3.4 Dirac delta and impulse response
    • 3.4.1 Rectangular pulse
    • 3.4.2 The delta function
    • 3.4.3 Impulse response
    • 3.4.4 Periodic functions
    • 3.4.5 Three-point beam bending
    • 3.4.6 Exercises
  • 3.5 Solving PDEs with the Laplace transform
    • 3.5.1 Exercises
• 4 Power series methods
  • 4.1 Power series
    • 4.1.1 Definition
    • 4.1.2 Radius of convergence
    • 4.1.3 Analytic functions
    • 4.1.4 Manipulating power series
    • 4.1.5 Power series for rational functions
    • 4.1.6 Analytic functions
    • 4.1.7 Exercises
  • 4.2 Series solutions of linear second order ODEs
    • 4.2.1 Exercises
  • 4.3 Singular points and the method of Frobenius
    • 4.3.1 Examples
    • 4.3.2 The method of Frobenius
    • 4.3.3 Bessel functions
    • 4.3.4 Exercises
• 5 Fourier series and PDEs
  • 5.1 Boundary value problems
    • 5.1.1 Boundary value problems
    • 5.1.2 Eigenvalue problems
    • 5.1.3 Orthogonality of eigenfunctions
    • 5.1.4 Fredholm alternative
    • 5.1.5 Application
    • 5.1.6 Exercises
  • 5.2 The trigonometric series
    • 5.2.1 Periodic functions and motivation
    • 5.2.2 The trigonometric series
    • 5.2.3 Exercises
  • 5.3 More on the Fourier series
    • 5.3.1 \(2L\)-periodic functions
    • 5.3.2 Convergence
    • 5.3.3 Differentiation and integration of Fourier series
    • 5.3.4 Rates of convergence and smoothness
    • 5.3.5 Exercises
  • 5.4 Sine and cosine series
    • 5.4.1 Odd and even periodic functions
    • 5.4.2 Sine and cosine series
    • 5.4.3 Application
    • 5.4.4 Exercises
  • 5.5 Applications of Fourier series
    • 5.5.1 Periodically forced oscillation
    • 5.5.2 Resonance
    • 5.5.3 Exercises
  • 5.6 PDEs, separation of variables, and the heat equation
    • 5.6.1 Heat on an insulated wire
    • 5.6.2 Separation of variables
    • 5.6.3 Insulated ends
    • 5.6.4 Exercises
  • 5.7 One-dimensional wave equation
    • 5.7.1 Exercises
  • 5.8 D’Alembert solution of the wave equation
    • 5.8.1 Change of variables
    • 5.8.2 D’Alembert’s formula
    • 5.8.3 Another way to solve for the side conditions
    • 5.8.4 Some remarks
    • 5.8.5 Exercises
  • 5.9 Steady state temperature and the Laplacian
    • 5.9.1 Exercises
  • 5.10 Dirichlet problem in the circle and the Poisson kernel
    • 5.10.1 Laplace in polar coordinates
    • 5.10.2 Series solution
    • 5.10.3 Poisson kernel
    • 5.10.4 Exercises
• 6 More on eigenvalue problems
  • 6.1 Sturm–Liouville problems
    • 6.1.1 Boundary value problems
    • 6.1.2 Orthogonality
    • 6.1.3 Fredholm alternative
    • 6.1.4 Eigenfunction series
    • 6.1.5 Exercises
  • 6.2 Higher order eigenvalue problems
    • 6.2.1 Exercises
  • 6.3 Steady periodic solutions
    • 6.3.1 Forced vibrating string
    • 6.3.2 Underground temperature oscillations
    • 6.3.3 Exercises
• 7 Systems of ODEs
  • 7.1 Introduction to systems of ODEs
    • 7.1.1 Systems
    • 7.1.2 Applications
    • 7.1.3 Changing to first order
    • 7.1.4 Autonomous systems and vector fields
    • 7.1.5 Picard’s theorem
    • 7.1.6 Exercises
  • 7.2 Matrices and linear systems
    • 7.2.1 Matrices and vectors
    • 7.2.2 Matrix multiplication
    • 7.2.3 The determinant
    • 7.2.4 Solving linear systems
    • 7.2.5 Computing the inverse
    • 7.2.6 Exercises
  • 7.3 Linear systems of ODEs
    • 7.3.1 Exercises
  • 7.4 Eigenvalue method
    • 7.4.1 Eigenvalues and eigenvectors of a matrix
    • 7.4.2 The eigenvalue method with distinct real eigenvalues
    • 7.4.3 Complex eigenvalues
    • 7.4.4 Exercises
  • 7.5 Two-dimensional systems and their vector fields
    • 7.5.1 Exercises
  • 7.6 Second order systems and applications
    • 7.6.1 Undamped mass-spring systems
    • 7.6.2 Examples
    • 7.6.3 Forced oscillations
    • 7.6.4 Exercises
  • 7.7 Multiple eigenvalues
    • 7.7.1 Geometric multiplicity
    • 7.7.2 Defective eigenvalues
    • 7.7.3 Exercises
  • 7.8 Matrix exponentials
    • 7.8.1 Definition
    • 7.8.2 Simple cases
    • 7.8.3 General matrices
    • 7.8.4 Fundamental matrix solutions
    • 7.8.5 Approximations
    • 7.8.6 Exercises
  • 7.9 Nonhomogeneous systems
    • 7.9.1 First order constant coefficient
      • 7.9.1.1 Integrating factor
      • 7.9.1.2 Eigenvector decomposition
      • 7.9.1.3 Undetermined coefficients
    • 7.9.2 First order variable coefficient
      • 7.9.2.1 Variation of parameters
    • 7.9.3 Second order constant coefficients
      • 7.9.3.1 Undetermined coefficients
      • 7.9.3.2 Eigenvector decomposition
    • 7.9.4 Exercises
• Back Matter
  • A Linear algebra
    • A.1 Vectors, mappings, and matrices
      • A.1.1 Vectors and operations on vectors
      • A.1.2 Linear mappings and matrices
      • A.1.3 Exercises
    • A.2 Matrix algebra
      • A.2.1 One-by-one matrices
      • A.2.2 Matrix addition and scalar multiplication
      • A.2.3 Matrix multiplication
      • A.2.4 Some rules of matrix algebra
      • A.2.5 Inverse
      • A.2.6 Diagonal matrices
      • A.2.7 Transpose
      • A.2.8 Exercises
    • A.3 Elimination
      • A.3.1 Linear systems of equations
      • A.3.2 Row echelon form and elementary operations
      • A.3.3 Non-unique solutions and inconsistent systems
      • A.3.4 Linear independence and rank
      • A.3.5 Computing the inverse
      • A.3.6 Exercises
    • A.4 Subspaces, dimension, and the kernel
      • A.4.1 Subspaces, basis, and dimension
      • A.4.2 Kernel
      • A.4.3 Exercises
    • A.5 Inner product and projections
      • A.5.1 Inner product and orthogonality
      • A.5.2 Orthogonal projection
      • A.5.3 Orthogonal basis
      • A.5.4 The Gram–Schmidt process
      • A.5.5 Exercises
    • A.6 Determinant
      • A.6.1 Exercises
  • B Table of Laplace Transforms
  • Further Reading
  • Index

Section 0.3 Classification of differential equations

There are many types of differential equations, and we classify them into different categories based on their properties. Let us quickly go over the most basic classification. We already saw the distinction between ordinary and partial differential equations:

• Ordinary differential equations or (ODE) are equations where the derivatives are taken with respect to only one variable. That is, there is only one independent variable.
• Partial differential equations or (PDE) are equations that depend on partial derivatives of several variables. That is, there are several independent variables.

Let us see some examples of ordinary differential equations: \begin{equation*} \begin{aligned} & \frac{d y}{dt} = ky , & & \text{(Exponential growth)} \\ & \frac{d y}{dt} = k(A-y) , & & \text{(Newton's law of cooling)} \\ & m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = f(t) . & & \text{(Mechanical vibrations)} \end{aligned} \end{equation*} And of partial differential equations: \begin{equation*} \begin{aligned} & \frac{\partial y}{\partial t} + c \frac{\partial y}{\partial x} = 0 , & & \text{(Transport equation)} \\ & \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} , & & \text{(Heat equation)} \\ & \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} . & & \text{(Wave equation in 2 dimensions)} \end{aligned} \end{equation*} If there are several equations working together, we have a so-called system of differential equations. For example, \begin{equation*} y' = x , \qquad x' = y \end{equation*} is a simple system of ordinary differential equations. Maxwell’s equations for electromagnetics, \begin{equation*} \begin{aligned} & \nabla \cdot \vec{D} = \rho, & & \nabla \cdot \vec{B} = 0 , \\ & \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, & & \nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t} , \end{aligned} \end{equation*} are a system of partial differential equations. The divergence operator \(\nabla \cdot\) and the curl operator \(\nabla \times\) can be written out in partial derivatives of the functions involved in the \(x\text{,}\) \(y\text{,}\) and \(z\) variables. The next bit of information is the order of the equation (or system). The order is simply the order of the largest derivative that appears. If the highest derivative that appears is the first derivative, the equation is of first order. If the highest derivative that appears is the second derivative, then the equation is of second order. For example, Newton’s law of cooling above is a first order equation, while the mechanical vibrations equation is a second order equation. The equation governing transversal vibrations in a beam, \begin{equation*} a^4 \frac{\partial^4 y}{\partial x^4} + \frac{\partial^2 y}{\partial t^2} = 0, \end{equation*} is a fourth order partial differential equation. It is fourth order as at least one derivative is the fourth derivative. It does not matter that the derivative in \(t\) is only of second order. In the first chapter, we will start attacking first order ordinary differential equations, that is, equations of the form \(\frac{dy}{dx} = f(x,y)\text{.}\) In general, lower order equations are easier to work with and have simpler behavior, which is why we start with them. We also distinguish how the dependent variables appear in the equation (or system). In particular, we say an equation is linear if the dependent variable (or variables) and their derivatives appear linearly, that is only as first powers, they are not multiplied together, and no other functions of the dependent variables appear. In other words, the equation is a sum of terms, where each term is some function of the independent variables or some function of the independent variables multiplied by a dependent variable or its derivative. Otherwise, the equation is called nonlinear. For example, an ordinary differential equation is linear if it can be put into the form \begin{equation} a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_{1}(x) \frac{dy}{dx} + a_{0}(x) y = b(x) .\tag{3} \end{equation} The functions \(a_0\text{,}\) \(a_1\text{,}\) ..., \(a_n\) are called the coefficients. The equation is allowed to depend arbitrarily on the independent variable. So \begin{equation} e^x \frac{d^2 y}{dx^2} + \sin(x) \frac{d y}{dx} + x^2 y = \frac{1}{x}\tag{4} \end{equation} is still a linear equation as \(y\) and its derivatives only appear linearly. All the equations and systems above as examples are linear. It may not be immediately obvious for Maxwell’s equations unless you write out the divergence and curl in terms of partial derivatives. Let us see some nonlinear equations. For example Burger’s equation, \begin{equation*} \frac{\partial y}{\partial t} + y \frac{\partial y}{\partial x} = \nu \frac{\partial^2 y}{\partial x^2} , \end{equation*} is a nonlinear second order partial differential equation. It is nonlinear because \(y\) and \(\frac{\partial y}{\partial x}\) are multiplied together. The equation \begin{equation} \frac{dx}{dt} = x^2\tag{5} \end{equation} is a nonlinear first order differential equation as there is a second power of the dependent variable \(x\text{.}\) A linear equation may further be called homogeneous if all terms depend on the dependent variable. That is, if no term is a function of the independent variables alone. Otherwise, the equation is called nonhomogeneous or inhomogeneous. For example, the exponential growth equation, the wave equation, or the transport equation above are homogeneous. The mechanical vibrations equation above is nonhomogeneous as long as \(f(t)\) is not the zero function. Similarly, if the ambient temperature \(A\) is nonzero, Newton’s law of cooling is nonhomogeneous. A homogeneous linear ODE can be put into the form \begin{equation*} a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_{1}(x) \frac{dy}{dx} + a_{0}(x) y = 0 . \end{equation*} Compare to (3) and notice there is no function \(b(x)\text{.}\) If the coefficients of a linear equation are actually constant functions, then the equation is said to have constant coefficients. The coefficients are the functions multiplying the dependent variable(s) or one of its derivatives, not the function \(b(x)\) standing alone. A constant coefficient nonhomogeneous ODE is an equation of the form \begin{equation*} a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_{1} \frac{dy}{dx} + a_{0} y = b(x) , \end{equation*} where \(a_0, a_1, \ldots, a_n\) are all constants, but \(b\) may depend on the independent variable \(x\text{.}\) The mechanical vibrations equation above is a constant coefficient nonhomogeneous second order ODE. The same nomenclature applies to PDEs, so the transport equation, heat equation and wave equation are all examples of constant coefficient linear PDEs. Finally, an equation (or system) is called autonomous if the equation does not depend on the independent variable. For autonomous ordinary differential equations, the independent variable is then thought of as time. Autonomous equation means an equation that does not change with time. For example, Newton’s law of cooling is autonomous, so is equation (5). On the other hand, mechanical vibrations or (4) are not autonomous.

Subsection 0.3.1 Exercises

Exercise0.3.1.

Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?

1. \(\displaystyle \displaystyle \sin(t) \frac{d^2 x}{dt^2} + \cos(t) x = t^2\)
2. \(\displaystyle \displaystyle \frac{\partial u}{\partial x} + 3 \frac{\partial u}{\partial y} = xy\)
3. \(\displaystyle \displaystyle y''+3y+5x=0, \quad x''+x-y=0\)
4. \(\displaystyle \displaystyle \frac{\partial^2 u}{\partial t^2} + u\frac{\partial^2 u}{\partial s^2} = 0\)
5. \(\displaystyle \displaystyle x''+tx^2=t\)
6. \(\displaystyle \displaystyle \frac{d^4 x}{dt^4} = 0\)

Answer.a) ODE, equation, second order, linear, non homogeneous, non constant coefficient, non autonomous. b) PDE, Equation, first order, linear, non homogeneous, constant coefficient. c) ODE, system, second order, linear, homogeneous, constant coefficient, autonomous. Note: independent variable is t here. d) PDE, equation, second order, non linear. e) ODE, equation, second order, non linear, non autonomous. f) ODE, equation, fourth order, linear, homogeneous, constant coefficient, autonomous.

Exercise0.3.2.

If \(\vec{u} = (u_1,u_2,u_3)\) is a vector, we have the divergence \(\nabla \cdot \vec{u} = \frac{\partial u_1}{\partial x} + \frac{\partial u_2}{\partial y} + \frac{\partial u_3}{\partial z}\) and curl \(\nabla \times \vec{u} = \Bigl( \frac{\partial u_3}{\partial y} - \frac{\partial u_2}{\partial z} , ~ \frac{\partial u_1}{\partial z} - \frac{\partial u_3}{\partial x} , ~ \frac{\partial u_2}{\partial x} - \frac{\partial u_1}{\partial y} \Bigr)\text{.}\) Notice that curl of a vector is still a vector. Write out Maxwell’s equations in terms of partial derivatives and classify the system. Answer. \begin{equation*} \begin{aligned} \frac{\partial D_1}{\partial x}+\frac{\partial D_2}{\partial y}+\frac{\partial D_3}{\partial z}=\rho, &\quad \frac{\partial B_1}{\partial x}+\frac{\partial B_2}{\partial y}+\frac{\partial B_3}{\partial z}=0 \\ \frac{\partial E_3}{\partial y}-\frac{\partial E_2}{\partial z}=-\frac{\partial B_1}{\partial t}, &\quad \frac{\partial H_3}{\partial y}-\frac{\partial H_2}{\partial z}=\frac{\partial D_1}{\partial t}+J_1 \\ \frac{\partial E_1}{\partial z}-\frac{\partial E_3}{\partial x}=-\frac{\partial B_2}{\partial t}, &\quad \frac{\partial H_1}{\partial z}-\frac{\partial H_3}{\partial x}=\frac{\partial D_2}{\partial t}+J_2 \\ \frac{\partial E_2}{\partial x}-\frac{\partial E_1}{\partial y}=-\frac{\partial B_3}{\partial t}, &\quad \frac{\partial H_2}{\partial x}-\frac{\partial H_1}{\partial y}=\frac{\partial D_3}{\partial t}+J_3 \end{aligned} \end{equation*} PDE, system, first order, linear, nonhomogeneous, constant coefficient, autonomous.

Exercise0.3.3.

Suppose \(F\) is a linear function, that is, \(F(x,y) = ax+by\) for constants \(a\) and \(b\text{.}\) What is the classification of equations of the form \(F(y',y) = 0\text{.}\) Answer.ODE, linear, homogeneous, constant coefficient, autonomous.

Exercise0.3.4.

Write down an explicit example of a third order, linear, nonconstant coefficient, nonautonomous, nonhomogeneous system of two ODE such that every derivative that could appear, does appear. Answer.An example would be \(y'''+xy''+y'+y=3x^2\text{.}\)

Exercise0.3.5.

Classify the following equations. Are they ODE or PDE? Is it an equation or a system? What is the order? Is it linear or nonlinear, and if it is linear, is it homogeneous, constant coefficient? If it is an ODE, is it autonomous?

1. \(\displaystyle \displaystyle \frac{\partial^2 v}{\partial x^2} + 3 \frac{\partial^2 v}{\partial y^2} = \sin(x)\)
2. \(\displaystyle \displaystyle \frac{d x}{dt} + \cos(t) x = t^2+t+1\)
3. \(\displaystyle \displaystyle \frac{d^7 F}{dx^7} = 3F(x)\)
4. \(\displaystyle \displaystyle y''+8y'=1\)
5. \(\displaystyle \displaystyle x''+tyx'=0, \quad y''+txy = 0\)
6. \(\displaystyle \displaystyle \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial s^2} + u^2\)

Answer.a) PDE, equation, second order, linear, nonhomogeneous, constant coefficient. b) ODE, equation, first order, linear, nonhomogeneous, not constant coefficient, not autonomous. c) ODE, equation, seventh order, linear, homogeneous, constant coefficient, autonomous. d) ODE, equation, second order, linear, nonhomogeneous, constant coefficient, autonomous. e) ODE, system, second order, nonlinear. f) PDE, equation, second order, nonlinear.

Exercise0.3.6.

Write down the general zeroth order linear ordinary differential equation. Write down the general solution. Answer.equation: \(a(x) y = b(x)\text{,}\) solution: \(y = \frac{b(x)}{a(x)}\text{.}\)

Exercise0.3.7.

For which \(k\) is \(\frac{dx}{dt}+x^k = t^{k+2}\) linear. Hint: there are two answers. Answer. \(k=0\) or \(k=1\) <Prev^TopNext> For a higher quality printout use the PDF version: https://www.jirka.org/diffyqs/diffyqs.pdf