In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. A simple example is Newton's second law of motion, which leads to the differential equation. for the motion of a particle of mass m.Nov 16, 2022 ... In this section we solve linear first order differential equations, i.e. differential equations in the form y' + p(t) y = g(t).Section 2.5 : Substitutions. In the previous section we looked at Bernoulli Equations and saw that in order to solve them we needed to use the substitution \(v = {y^{1 - n}}\). Upon using this substitution, we were able to convert the differential equation into a form that we could deal with (linear in this case).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 …Earlier, we studied an application of a first-order differential equation that involved solving for the velocity of an object. In particular, if a ball is thrown upward with an initial velocity of \( v_0\) ft/s, then an initial-value problem that describes the velocity of the ball after \( t\) seconds is given byNov 16, 2022 · A second order, linear nonhomogeneous differential equation is. y′′ +p(t)y′ +q(t)y = g(t) (1) (1) y ″ + p ( t) y ′ + q ( t) y = g ( t) where g(t) g ( t) is a non-zero function. Note that we didn’t go with constant coefficients here because everything that we’re going to do in this section doesn’t require it. Also, we’re using ... Ordinary Differential Equations An ordinary differential equation (or ODE) is an equation involving derivatives of an unknown quantity with respect to a single variable. More precisely, suppose j;n2 N, Eis a Euclidean space, and FW dom.F/ R nC 1copies ‚ …„ ƒ E E! Rj: (1.1) Then an nth order ordinary differential equation is an equation ... 4 days ago · A linear ordinary differential equation of order is said to be homogeneous if it is of the form. (1) where , i.e., if all the terms are proportional to a derivative of (or itself) and there is no term that contains a function of alone. However, there is also another entirely different meaning for a first-order ordinary differential equation. Exercise 1.E. 1.1.11. A dropped ball accelerates downwards at a constant rate 9.8 meters per second squared. Set up the differential equation for the height above ground h in meters. Then supposing h(0) = 100 meters, how long does it …Ordinary Differential Equations 2: First Order Differential Equations 2.8: Theory of Existence and Uniqueness ... It is easier to prove that the integral equation has a unique solution, then it is to show that the original differential equation has a unique solution. The strategy to find a solution is the following. First guess at a solution ...Ordinary Differential Equations (ODEs for short) come up whenever you have an exact relationship between variables and their rates. Therefore you.In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form. where is a real number. Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [3] [4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named.ordinary differential equation (ODE), in mathematics, an equation relating a function f of one variable to its derivatives. (The adjective ordinary here refers to those …Michigan State UniversityIn this chapter we will look at solving first order differential equations. The most general first order differential equation can be written as, dy dt = f (y,t) (1) (1) d y d t = f ( y, t) As we will see in this chapter there is no general formula for the solution to (1) (1). What we will do instead is look at several special cases and see how ...An ordinary differential equation (ODE) is a differential equation in mathematics that has one or more functions of one independent variable and its derivatives ...Unit 1: First order differential equations. Intro to differential equations Slope fields Euler's Method Separable equations. Exponential models Logistic models Exact equations and integrating factors Homogeneous equations. The main equations studied in the course are driven first and second order constant coefficient linear ordinary differential equations and 2x2 systems. For these equations students will be able to: Use known DE types to model and understand situations involving exponential growth or decay and second order physical systems such as driven spring ...Section 3.4 : Repeated Roots. In this section we will be looking at the last case for the constant coefficient, linear, homogeneous second order differential equations. In this case we want solutions to. ay′′ +by′ +cy = 0 a y ″ + b y ′ + c y = 0. where solutions to the characteristic equation. ar2+br +c = 0 a r 2 + b r + c = 0.An Ordinary Differential Equation (ODE)is a differential equation containing (ordinary) derivatives of a function y = f(x) which has only one independent variable x. Note that “Ordinary” derivatives are the derivatives presented in these concepts. A Partial Differential Equation (PDE) is a differential equation containing derivatives …Discretization of ODE system. I am fairly new to the discretization of ODE systems (indeed a good reference would be helpful). I have a system of ODEs that basically looks like this. dx(t) dt dv(t) dt = v(t) = a(t,xt,vt) d x ( t) d t = v ( t) d v ( t) d t = a ( t, x t, v t) How do I discretize this and , given a discretization, how do I know if ...An ordinary differential equation (ODE) is a type of differential equation that involves a single independent variable and its derivatives (e.g., x, dx/dt). It describes how a function changes as time passes or as other variable changes, such as temperature or pressure. These equations are used to model physical phenomena such as gravity ...Ordinary Di erential Equation De nition Let I be an open interval of R. A k-th order ordinary di erential equation of an unknown function y : I !R is of the form F y(k);y(k 1);:::y0(x);y(x);x = 0; (3.1) for each x 2I, where F : Rk+1 I !R is a given map such that F depends on the k-th order derivative y and is independent of (k + j)-thThe main equations studied in the course are driven first and second order constant coefficient linear ordinary differential equations and 2x2 systems. For these equations students will be able to: Use known DE types to model and understand situations involving exponential growth or decay and second order physical systems such as driven spring ...Nov 12, 2006 · Ince, Ordinary Differential Equations, was published in 1926. It manages to pack a lot of good material into 528 pages. (With appendices it is 547 pages, but they are no longer relevant.) I have used Ince for several decades as a handy reference for Differential Equations. MSC: Primary 34; 37;. This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate ...Free Series Solutions to Differential Equations Calculator - find series solutions to differential equations step by step ... ode-series-solutions-calculator. en. Related Symbolab blog posts. Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE. Last post, we talked about linear first order differential ...Ordinary Differential Equations is based on the author's lecture notes from courses on ODEs taught to advanced undergraduate and graduate students in mathematics, physics, and engineering. The book, which remains as useful today as when it was first published, includes an excellent selection of exercises varying in difficulty from routine ...Stiff equation. In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms ... As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in n - dimensional Euclidean space R n and driven by an m -dimensional Brownian motion B ; the ... First Order Linear. First Order Linear Differential Equations are of this type: dy dx + P (x)y = Q (x) Where P (x) and Q (x) are functions of x. They are "First Order" when there is only dy dx (not d2y dx2 or d3y dx3 , etc.) Note: a non-linear differential equation is often hard to solve, but we can sometimes approximate it with a linear ... Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in …3.7: Uniqueness and Existence for Second Order Differential Equations. if p(t) p ( t) and g(t) g ( t) are continuous on [a, b] [ a, b], then there exists a unique solution on the interval [a, b] [ a, b]. We can ask the same questions of second order linear differential equations. We need to first make a few comments.1.1 Ordinary Differential Equation (ODE) An equation involving the derivatives of an unknown function y of a single variable x over an interval x ∈ (I). More clearly and precisely speaking, a well defined ODE must the following features: It can be written in the form: F[x,y,y′,y′′,···,yn] = 0; (1.1) Ordinary Differential Equations ... The Ordinary Differential Equation (ODE) solvers in MATLAB® solve initial value problems with a variety of properties. The ...A carefully revised edition of the well-respected ODE text, whose unique treatment provides a smooth transition to critical understanding of proofs of basic ...I am reading Wikipedia's entry on Flow and it is not clear the distinction between solution of an ODE and the flow of an ODE. In particular it is clearly written $φ(x_0,t) = x(t)$, then what is the ... It can be associated for example to a stochastic differential equation, a delay equation, a partial differential equation, or even be ...The differential equation solvers in MATLAB ® cover a range of uses in engineering and science. There are solvers for ordinary differential equations posed as either initial value problems or boundary value problems, delay differential equations, and partial differential equations. Additionally, there are functions to integrate functional ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Dec 26, 2018 · About the Book. This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take. This book consists of 10 chapters, and the course is 12 weeks long. Sep 7, 2022 · Add the general solution to the complementary equation and the particular solution found in step 3 to obtain the general solution to the nonhomogeneous equation. Example 17.2.5: Using the Method of Variation of Parameters. Find the general solution to the following differential equations. y″ − 2y′ + y = et t2. An ordinary differential equation (ODE) is a mathematical equation involving a single independent variable and one or more derivatives, while a partial differential equation (PDE) involves multiple independent variables and partial derivatives. Example 1. Solve the ordinary differential equation (ODE) dx dt = 5x − 3 d x d t = 5 x − 3. for x(t) x ( t). Solution: Using the shortcut method outlined in the introduction to ODEs, we multiply through by dt d t and divide through by 5x − 3 5 x − 3 : dx 5x − 3 = dt. d x 5 x − 3 = d t. We integrate both sides.Differential equations are important because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that ...An ODE is referred to as a neural ordinary differential equation (neuralODE) when it is used to describe the dynamics of a neural network. All the ODEs discussed in this paper are considered neuralODEs. A general neuralODE can be described as. $$\begin {aligned} {\dot {y}} = f (y,x) \end {aligned}$$.Ordinary Differential Equations: Classification of ODEs Classification of ODEs Order. The order of an ODE is the order of the highest derivative appearing in the equation. For …Ordinary Differential Equations 2: First Order Differential Equations 2.8: Theory of Existence and Uniqueness ... It is easier to prove that the integral equation has a unique solution, then it is to show that the original differential equation has a unique solution. The strategy to find a solution is the following. First guess at a solution ...Nov 30, 2021 · DEFINITION 1: ORDINARY DIFFERENTIAL EQUATIONS. An ordinary differential equation (ODE) is an equation for a function of one variable that involves (‘’ordinary”) derivatives of the function (and, possibly, known functions of the same variable). We give several examples below. d2x dt2 + ω2x = 0. d 2 x d t 2 + ω 2 x = 0. Learn the basics of solving ordinary differential equations in MATLAB. Use MATLAB ODE solvers to find solutions to ordinary differential equations that describe phenomena ranging from population dynamics to the evolution of the universe. Ordinary Di erential Equation De nition Let I be an open interval of R. A k-th order ordinary di erential equation of an unknown function y : I !R is of the form F y(k);y(k 1);:::y0(x);y(x);x = 0; (3.1) for each x 2I, where F : Rk+1 I !R is a given map such that F depends on the k-th order derivative y and is independent of (k + j)-th常微分方程式 （じょうびぶんほうていしき、 英: ordinary differential equation, O.D.E. ）とは、 微分方程式 の一種で、 未知関数 が本質的にただ一つの変数を持つものである場合をいう。. すなわち、変数 t の未知関数 x(t) に対して、（既知の）関数 F を用いて. と ... Boyce and DiPrima, Elementary Differential Equations, 9th edition (Wiley, 2009, ISBN 978-0-470-03940-3), Chapters 2, 3, 5 and 6 (but not necessarily in that order). Note that you are expected to bring the text to class each day (except on test days), so that we can refer to diagrams such as those which appear on pp. 9, 37 or 43 Section 3.3 : Complex Roots. In this section we will be looking at solutions to the differential equation. ay′′ +by′ +cy = 0 a y ″ + b y ′ + c y = 0. in which roots of the characteristic equation, ar2+br +c = 0 a r 2 + b r + c = 0. are complex roots in the form r1,2 = λ±μi r 1, 2 = λ ± μ i. Now, recall that we arrived at the ...Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. 1.1 Ordinary Differential Equation (ODE) An equation involving the derivatives of an unknown function y of a single variable x over an interval x ∈ (I). More clearly and precisely speaking, a well defined ODE must the following features: It can be written in the form: F[x,y,y′,y′′,···,yn] = 0; (1.1) Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Jun 10, 2023 · Equations of the form dy dx = f(Ax + By + C) Theorem 2.4.3. The substitution u = Ax + By + C will make equations of the form dy dx = f(Ax + By + C) separable. Proof. Consider a differential equation of the form 2.4.5. Let u = Ax + By + C. Taking the derivative with respect to x we get du dx = A + Bdy dx. Let’s take a look at an example. Example 1 Determine the Taylor series for f (x) = ex f ( x) = e x about x = 0 x = 0 . Of course, it’s often easier to find the Taylor series about x = 0 x = 0 but we don’t always do that. Example 2 Determine the Taylor series for f (x) = ex f ( x) = e x about x = −4 x = − 4 .Ordinary Differential Equations (ODEs) include a function of a single variable and its derivatives. The general form of a first-order ODE is $$ F\left(x,y,y^{\prime}\right)=0, $$ where $$$ y^{\prime} $$$ is the first derivative of $$$ y $$$ with respect to $$$ x $$$.Jun 26, 2023 · Here is a set of notes used by Paul Dawkins to teach his Differential Equations course at Lamar University. Included are most of the standard topics in 1st and 2nd order differential equations, Laplace transforms, systems of differential eqauations, series solutions as well as a brief introduction to boundary value problems, Fourier series and partial differntial equations. Sorted by: 8. A differential form is an expression ω = adx + bdy ω = a d x + b d y where dx, dy d x, d y are linear functionals on the tangent space. That is, if v = (v1,v2) v = ( v 1, v 2) is a direction, then dx(v) =v1 d x ( v) = v 1 and dy(v) =v2 d y ( v) = v 2. The equation ω = 0 ω = 0 describes a line 0 =ω(v) = av1 + bv2 0 = ω ( v ...Ordinary Differential Equations ... The Ordinary Differential Equation (ODE) solvers in MATLAB® solve initial value problems with a variety of properties. The ...dx dt = t2, d x d t = t 2, we can quickly solve it by integration. This equation is so simple because the left hand side is just a derivative with respect to t t and the right hand side is just a function of t t. We can solve by integrating both sides with respect to t t to get that x(t) = t3 3 + C x ( t) = t 3 3 + C .∆f. ∆x . A differential equation is an equation which contains derivatives and the goal is usually to solve it. ie To find the function (for engineers ...3.7: Uniqueness and Existence for Second Order Differential Equations. if p(t) p ( t) and g(t) g ( t) are continuous on [a, b] [ a, b], then there exists a unique solution on the interval [a, b] [ a, b]. We can ask the same questions of second order linear differential equations. We need to first make a few comments.May 19, 2022 ... The notation of the differential equations depends on the order of the functions such as first-order ODE has a notation dy/dx or y'(x), the ...Dec 26, 2018 · About the Book. This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take. This book consists of 10 chapters, and the course is 12 weeks long. May 28, 2023 · 4) You can determine the behavior of all first-order differential equations using directional fields or Euler’s method. Solution: \(\displaystyle T\) For the following problems, find the general solution to the differential equations. An Ordinary Differential Equation (ODE)is a differential equation containing (ordinary) derivatives of a function y = f(x) which has only one independent variable x. Note that “Ordinary” derivatives are the derivatives presented in these concepts. A Partial Differential Equation (PDE) is a differential equation containing derivatives …3. Formula sheet & practice exam with solutions ( PDF ) ( PDF ) ( PDF ) Final. Practice final exam ( PDF) and solutions ( PDF ) ( PDF ) [Solution not available] This section provides practice exams, exams, and solutions. High performance ordinary differential equation (ODE) and differential-algebraic equation (DAE) solvers, including neural ordinary differential equations (neural ODEs) and scientific machine learning (SciML) - SciML/OrdinaryDiffEq.jlRepeated Roots – In this section we discuss the solution to homogeneous, linear, second order differential equations, ay′′ +by′ +cy = 0 a y ″ + b y ′ + c y = 0, in which the roots of the characteristic polynomial, ar2 +br+c = 0 a r 2 + b r + c = 0, are repeated, i.e. double, roots. We will use reduction of order to derive the second ...This set of Ordinary Differential Equations Multiple Choice Questions & Answers focuses on “Solution of DE With Constant Coefficients using the Laplace Transform”. 1. While solving the ordinary differential equation using unilateral laplace transform, we consider the initial conditions of the system. a) True. b) False.An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given ...Description. ode solves explicit Ordinary Different Equations defined by:. It is an interface to various solvers, in particular to ODEPACK. In this help, we only describe the use of ode for standard explicit ODE systems.. The simplest call of ode is: y = ode(y0,t0,t,f) where y0 is the vector of initial conditions, t0 is the initial time, t is the vector of times at which the …Overview of ODEs. There are four major areas in the study of ordinary differential equations that are of interest in pure and applied science. Exact solutions, which are closed-form or implicit analytical expressions that satisfy the given problem. Numerical solutions, which are available for a wider class of problems, but are typically only ... Ordinary differential equations are much more understood and are easier to solve than partial differential equations, equations relating functions of more than one variable. We do not solve partial differential equations in this article because the methods for solving these types of equations are most often specific to the equation. [1]For the numerical solution of ODEs with scipy, see scipy.integrate.solve_ivp, scipy.integrate.odeint or scipy.integrate.ode. Some examples are given in the SciPy Cookbook (scroll down to the section on "Ordinary Differential Equations").Feb 1, 2024 ... @StephenLuttrell According to the discussion of Frobenius method in en.wikipedia.org/wiki/Frobenius_method, d = 0 is required to apply it to the ...Free Series Solutions to Differential Equations Calculator - find series solutions to differential equations step by step ... ode-series-solutions-calculator. en. Related Symbolab blog posts. Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE. Last post, we talked about linear first order differential ...3. Formula sheet & practice exam with solutions ( PDF ) ( PDF ) ( PDF ) Final. Practice final exam ( PDF) and solutions ( PDF ) ( PDF ) [Solution not available] This section provides practice exams, exams, and solutions. y′+p(t)y=f(t). ... Note: When the coefficient of the first derivative is one in the first order non-homogeneous linear differential equation as in the above ...Ordinary Differential Equations: Classification of ODEs Classification of ODEs Order. The order of an ODE is the order of the highest derivative appearing in the equation. For example, the following equation (Newton’s equation) is a second-order ODE: while the beam equation is a fourth-order ODE: Linear vs. Nonlinear
Nonlinear equations. The power series method can be applied to certain nonlinear differential equations, though with less flexibility. A very large class of nonlinear equations can be solved analytically by using the Parker–Sochacki method.Since the Parker–Sochacki method involves an expansion of the original system of ordinary differential equations …. Ercari
Example 1. Solve the ordinary differential equation (ODE) dx dt = 5x − 3 d x d t = 5 x − 3. for x(t) x ( t). Solution: Using the shortcut method outlined in the introduction to ODEs, we multiply through by dt d t and divide through by 5x − 3 5 x − 3 : dx 5x − 3 = dt. d x 5 x − 3 = d t. We integrate both sides. This course provides an introduction into ordinary (i.e. one-variable) differential equations, their analytical and numerical solution techniques and the ...The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable \(t\). We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain.Oct 24, 2023 ... Description · If f is a Scilab function, its syntax must be. ydot = f(t,y) · If f is a string, it is the name of a Fortran subroutine or a C ...This chapter covers ordinary differential equations with specified initial values, a subclass of differential equations problems called initial value problems. To reflect the importance of this class of problem, Python has a whole suite of functions to solve this kind of problem. By the end of this chapter, you should understand what ordinary ...Solver for Ordinary Differential Equations (ODE) Description. Solves the initial value problem for stiff or nonstiff systems of ordinary differential equations (ODE) in the form: dy/dt = f(t,y). The R function lsode provides an interface to the FORTRAN ODE solver of the same name, written by Alan C. Hindmarsh and Andrew H. Sherman.Ordinary differential equations (ODEs) and linear algebra are foundational postcalculus mathematics courses in the sciences. The goal of this text is to help ...Figure \(\PageIndex{1}\): The scheme for solving an ordinary differential equation using Laplace transforms. One transforms the initial value problem for \(y(t)\) and obtains an algebraic equation for \(Y(s)\). Solve for \(Y(s)\) and the inverse transform gives the solution to the initial value problem.A differential equation is an equation involving a function and its derivatives. It can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on whether or not partial derivatives are involved. Wolfram|Alpha can solve many problems under this important branch of mathematics, including ... Number Line · 2 y ′− y =4sin(3 t ) · ty ′+2 y = t− t +1 · y ′= e (2 x −4) · dr d θ = r θ · y ′+4 x y = x y · y ′+4 x y = x y, y (2)=−1 &mi...Basic Concepts – In this section give an in depth discussion on the process used to solve homogeneous, linear, second order differential equations, ay′′ +by′ +cy = 0 a y ″ + b y ′ + c y = 0. We derive the characteristic polynomial and discuss how the Principle of Superposition is used to get the general solution.Sep 7, 2022 · Second-order constant-coefficient differential equations can be used to model spring-mass systems. An examination of the forces on a spring-mass system results in a differential equation of the form \[mx″+bx′+kx=f(t), onumber \] where mm represents the mass, bb is the coefficient of the damping force, \(k\) is the spring constant, and \(f ... Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes …In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form. where is a real number. Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [3] [4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. Ordinary Differential Equations. Frank E. Harris, in Mathematics for Physical Science and Engineering, 2014 Abstract. This chapter deals with ordinary differential equations (ODEs). First-order ODEs that are separable, exact, or homogeneous in both variables are discussed, as are methods that use an integrating factor to make a linear ODE exact.They are distinct from ordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x because these may not all appear (i.e. some equations are algebraic); technically the distinction between an implicit ODE system [that may be rendered explicit] and a DAE system is that the ...Stiff equation. In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms ...Oct 18, 2018 · Exercise 8.1.1 8.1. 1. Verify that y = 2e3x − 2x − 2 y = 2 e 3 x − 2 x − 2 is a solution to the differential equation y' − 3y = 6x + 4. y ′ − 3 y = 6 x + 4. Hint. It is convenient to define characteristics of differential equations that make it easier to talk about them and categorize them. The differential equation solvers in MATLAB ® cover a range of uses in engineering and science. There are solvers for ordinary differential equations posed as either initial value problems or boundary value problems, delay differential equations, and partial differential equations. Additionally, there are functions to integrate functional ...4 days ago · A linear ordinary differential equation of order is said to be homogeneous if it is of the form. (1) where , i.e., if all the terms are proportional to a derivative of (or itself) and there is no term that contains a function of alone. However, there is also another entirely different meaning for a first-order ordinary differential equation. .