Preface

---

Differential equations were invented in the second part of the seventeenth century by I. Newton and G. Leibniz in order to describe some physical phenomena in the appropriate language. The term aequatio differentialis or differential equation was first used by Leibniz in 1676 to denote a relationship between the differentials dx and dy of two variables x and y. In contrast to algebraic equations, differential equations deal with classes of functions. This topic provides a fantastic tool for modeling situations in any field or industry.

Differential equations fall into two very broad categories, called ordinary differential equations (ODE for short) and partial differential equations (PDE). If the unknown function in the equation is a function of only one variable, the equation is called an ordinary differential equation. If the unknown function in the equation depends on more than one independent variable, the equation is called a partial differential equation. This chapter is devoted to some classes of of ordinary differential equation of order one, that is, the equation contains only the first derivative of an unknown function.

 

Before beginning to tackle problem formulation and solving differential equations, it is necessary to formulate some basic terminology. First, we reiterate the following notations used to represent a derivative of a function f(x) of one independent variable x:

• In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of f with respect to x is written \( {\text d}f / {\text d} x . \) The above expression is read as "the derivative of f with respect to x", "df by dx", or "df over dx". The second derivative is denoted as \( {\text d}^2 f / {\text d} x^2 . \)
• In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted \( f'(x) \) (read as "f prime of x") or fx′(x) (read as "f prime x of x"). The second derivative is denoted as \( f'' (x) = {\text d}^2 f / {\text d} x^2 . \)
• In Newton's notation, the derivative with respect to time variable t of a function y(t) is denoted by dot: \( \dot{y} = {\text d}y / {\text d} t . \) The second derivative is denoted as \( \ddot{y} = {\text d}^2 y / {\text d} t^2 . \)
• Leonhard Euler's notation uses a differential operator suggested by Louis François Antoine Arbogast, denoted as \( \texttt{D} = {\text d} / {\text d} x . \) The second power of the derivative operator is denoted as \( \texttt{D}^2 = {\text d}^2 / {\text d} x^2 . \)

A differential equation may contain three different types of quantities. The unknown function, for which the equation is to be solved, is called the dependent variable, and when considering ordinary differential equations, the dependent variable is a function of a single independent variable (which we will usually use letters either x or t, for time). In addition to the independent to the independent and dependent variables, a third type of variable, called a parameter, may appear in the equation. A parameter is a quantity that remains fixed in any specification of the problem, but can vary from problem to problem.

A study of differential equations initiated by Gottfried Wilhelm (von) Leibniz in 1676, originated from modeling mechanical problems that are usually written in differentials as

It is approximated by the equation Here x is the independent variable and y is the dependent variable. Division by Δx yields the equation It should be emphasized that the differential equation follows as an exact mathematical consequence, even though the initial equality is only an approximation.

We utilize the term “differential equations” for the various equations used in this course because they were originally applied to equations formulated in differentials, which deal with incremental changes. The terminology is derived from physics, where they were used to express physical laws such as those of motion, electricity, or magnetism. Differential equations were designed to express these laws such that the physical rules are fixed, yet the same equation could be applied in different contexts. For example, a differential equation of Newton's second law of motion that includes a derivative of velocity with respect to time can be used to find the incremental changes of velocity in a variety of situations. In this course, we adopt the historical convention of calling all equations involving differentials or derivatives, differential equations rather than derivative equations.

 

One important classification is based on whether the unknown function depends on a single independent variable or on several independent variables. This first part of the tutorial deals only with solutions depending on one variable, and they are said to satisfy an ordinary differential equation (abbreviated as ODE). When a differential equation is set for a function depending on several independent variables, we call such equation the partial differential equation (abbreviated as PDE). Example: Ordinary differential equation Example: Partial differential equations

 

A first-order differential equation is an equation of the form where F(x,y,p) is a real-valued function of three variables, and G(x,y,p,q) is a real-valued function of four variables. It is assumed that the derivative y' at least occurs explicitly on the left-hand side of the above equation with F. A solution to \( F(x,y, y' ) =0 \) is a function \( y = \phi (x) \) so that \( F\left( x,\phi (x) , \phi' (x) \right) \equiv 0 \) for all x in some open interval (𝑎,b). In particular, we require the solution function to be differentiable, and hence continuous on (𝑎,b). A second order differential equation is the equation of the form for some continuous function F of four variables.

We will usually deal with differential equations where the derivative is isolated instead of its general form F(x,y,y') = 0. If the function of three variables F(x,y,p) satisfies the conditions (F is continuously differentiable and the partial derivative F_p ≠ 0) of the implicit function theorem, then p can be expressed as a continuous function p = y' = f(x,y), and we can isolate the derivative.

We will also use dots to identify the derivatives with respect to time variable: \( \dot{y} = {\text d} y / {\text d} t , \quad \ddot{y} = {\text d}^2 y/ {\text d} t^2 . \)

An initial value problem for a first-order differential equation consists of the differential equation together with the specification of a value of the solution at a particular point. That is, initial value problems are of the general form:

where \( (x_0 , y_0 ) \) is a given point on the xy-plane. Example: Differential equation of distinct orders

 

Example: Linear and nonlinear differential equations

 

We will learn shortly that a differential equation usually has infinitely many solutions depending on arbitrary constants. Now we go in opposite direction and give some examples of the elimination of arbitrary constants by the formation of ordinary differential equations. Historically, the first differential equations were derived from the equation \( F(x,y,c_1, c_2 , \ldots , c_n ) =0 \) upon differentiating n times to eliminate constants c₁, … ,c_n. For example, if we have one constant, the equation can be solved for it: c = ψ(x,y). Then differentiation with respect to x yields the first order differential equation \( \psi_x + \psi_y \,y' =0 . \) Before 1774 when G. Lagrange introduced the word "general solution," the equation \( F(x,y,c_1, c_2 , \ldots , c_n ) =0 \) was called the primitive for the corresponding differential equation. Example: Ellipse Example: Parabolas Example: Trigonometric functions

 

A general first order differential equation where \( y' = {\text d}y/{\text d}x \) is the derivative of the unknown function y(x), may have a family of solutions depending on a parameter C: \( \phi (x,y ,C ) =0 . \) This is expected because the derivative operator annihilates any constant. We most likely do not know a formula for the solution ϕ and consider it as a general expression for representing a family of solutions. By specifying particular values for C, we obtain a particular solution to the given differential equation.

There is another approach to knock down a particular solution by specifying the value of unknown function y(x) at particular point. Such condition is called the initial condition: y(x₀) = y₀, for some specified values x₀ and y₀. Then the particular value of a constant C can be determined from the equation \( \phi (x_0 ,y_0 ,C ) =0 , \) subject that it can be solved.

Example: the initial value problem We will discuss the validity intervals for autonomous equation later in more details.

 

It turns out that some points on the plane may not be used to identify the initial condition for a differential equation. Solutions in a neighborhood of some points may exhibit a 'nasty' behavior, such as a cusp or similar feature, which qualify them as exceptional points. At such points, the corresponding initial value problem may have either no solution or many solutions. Therefore, such exceptional points are excluded from consideration and analysis, and we label them with a special term.

The initial conditions are usually not specified at a singular point because the corresponding initial value problem may have multiple solutions or may have no solution at all. Example: nonexistence of the IVP

It is important to note that in a situation where the numerator is nonzero but the denominator is zero, this does not qualify a singular point. This just indicates that the tangent is vertical because dy/dx = ∞, but the geometric interpretation remains meaningful. In this case, we can simply flip the differential equation from dy/dx to dx/dy, moving the zero to the numerator and the nonzero to the denominator. This will give a slope of 0 in the horizontal direction. Thus, there is merit in treating x and y symmetrically, so that the independent variable can be either x or y.

The same conclusion is suggested when differential equations are applied to practical problems. Nature has no cognizance of coordinate systems, which merely provide a framework for the mathematical modeling of an underlying reality. If a problem seems intractable when we insist on a solution y = ϕ(x), but easier when we allow x = ψ(y). It could mean that we have made an inappropriate choice of independent and dependent variables in the initial formulation. These remarks motivate the formulation of the equation in differentials where all variables are treated equally.

 

Nullclines are usually not solutions to the differential equation except the particular case when they are constants. Example: nullclines for the Riccati equation

The behavior of the integral curves depends strongly on the structure of nullclines around critical points, that is, the multiplicity of nullclines and the sign of slope function.

 

Classifying equilibrium solutions of autonomous differential equations \( {\text d}y/{\text d}t = f(y) \) includes the following steps.

• Draw a vertical line (the phase line) and make tick marks at equilibrium values.
• Between tick marks determine if the slope function f(y) is positive or negative.
• If f(y) is positive, then any solution in this region will be increasing.
• If f(y) is negative, then any solution in this region will be decreasing.

Make conclusion:

• Increasing below and decreasing above     ⇒    asymptotically stable.
• Decreasing below and increasing above     ⇒    unstable.
• Decreasing below and above OR increasing below and above     ⇒     semistable.

Example: equilibrium solutions Equilibrium solutions can be generalized by including not straight lines.

The next example will show that there exist some curves but not straight lines. In this case it can be seen that solutions approach some curve, but not straight line. However, it is not easy to find such curve, and you need to do some extra work.

Example: curved asymptotic solution

 

The Wolfram Language function DSolve finds symbolic solutions to differential equations. The Wolfram Language function NDSolve, on the other hand, is a general numerical differential equation solver. DSolve can handle the following types of equations:

• Ordinary Differential Equations (ODEs), in which there are two or more independent variables and one dependent variable. Finding exact symbolic solutions of ODEs is a difficult problem, but DSolve can solve many first-order ODEs and a limited number of the second-order ODEs found in standard reference books.
• Partial Differential Equations (PDEs), in which there are two or more independent variables and one dependent variable. Finding exact symbolic solutions of PDEs is a difficult problem, but DSolve can solve most first-order PDEs and a limited number of the second-order PDEs found in standard reference books.
• Differential Algebraic Equations (DAEs), in which some members of the system are differential equations and the others are purely algebraic, having no derivatives in them. As with PDEs, it is difficult to find exact solutions to DAEs, but DSolve can solve many examples of such systems that occur in applications. DSolve

 

Mathematica can be used to verify that known functions are solutions to different differential equations.

Example. To verify that the function \( y=e^{2\,x} \) is a solution of the differential equation y' - 2 y=0 type:

If you enter this syntax correctly, you should get two outputs. The first output should read \( e^{2\,x} . \) You should get the second output by plugging this potential solution into the differential equation on the third line, which gives a result of zero.

As another example, consider the exponential function multiplied by a constant, \( y=c\,e^{x^2} . \) We show that this function is a solution to the differential equation \( y' -2x\, y =0 \) independently of arbitrary constant c:

In this syntax, Exp[ ] is the exponential command. It is necessary to make sure that you use the correct brackets. If you do not use the correct type of brackets then the command will not work. We also define a function of a given variable by placing that variable in brackets next to the function. To take derivatives you use '. This means "y-prime" in this context. When you enter this syntax, you once again get two outputs. The first outputs tells you what the function you are defining as y[x_] is. The second output tells you that this statement is true and cannot be simplified.