1.1 Introduction

A. Definitions

Differential equations (DEs) are mathematical equations that describe the relationship between a function and its derivatives, either ordinary derivatives or partial derivatives. In its simplest form, it describes the rate at which a quantity changes in terms of the quantity itself and its derivatives. Differential equations are powerful tools in mathematics and science as they enable the modeling of a wide range of real-world phenomena across various disciplines, including physics, engineering, biology, economics, and many others. Here are a few examples of differential equations.

• Basic population growth: [asciimath](dP)/(dt)=aP[/asciimath]
• Basic radioactive decay: [asciimath](dQ)/dt=-kQ[/asciimath]
• Newton’s laws of cooling: [asciimath](dT)/(dt)=-k(T-T_m)[/asciimath]
• Second Newton’s law of motion: [asciimath](d^2 y)/(dx^2 )=-g[/asciimath]
• RL circuits: [asciimath]L (dI)/(dt)+RI=E(t)[/asciimath]
• RLC circuits: [asciimath]L (d^2 I)/(dt^2 )+RI+1/C I=(dE)/dt[/asciimath]
• Heat equation: [asciimath](del u)/(del t) =beta (del^2 u)/(del x^2)[/asciimath]

B. Order of Differential Equations

The order of a differential equation is the order of the highest derivative that appears in the equation. For example, if the highest derivative is a second derivative, the equation is of second order. Here are a few examples:

 [asciimath]dy/dx+3x^2=0[/asciimath]    (First Order)

 [asciimath](d^2y)/dt^2 -2dy/dt + y= -5x[/asciimath]    (Second Order)

 [asciimath]x^2y'''+xy=cosx[/asciimath]   (Third Order)

 [asciimath](del u)/(del t)=5(del^2 u)/(del t^2)[/asciimath]   (Second Order)

The order of a differential equation often determines the methods used to solve it. The order of a differential equation is independent of the type of derivatives involved, whether they are ordinary or partial derivatives.

Throughout this book, our focus will primarily be on first- and second-order differential equations. As you’ll discover, the methods used to solve second-order differential equations can often be easily extended to tackle higher-order equations.

C. Ordinary and Partial Differential Equations

A Partial Differential Equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables.

In this textbook, our primary focus will be on ordinary differential equations, which involve functions of a single variable. We will only delve into partial differential equations in the final chapter.

D. Linear and nonlinear Differential Equations

A linear differential equation is one in which the dependent variable [asciimath]y[/asciimath] and its derivatives appear with the first power, are not multiplied together, and are not arguments of another function, e.g., [asciimath]sin(y)[/asciimath] or [asciimath]ln(y')[/asciimath]. The general form of a linear differential equation is

 [asciimath]a_n(x) (d^ny)/dx^n+a_(n-1)(x) (d^(n-1) y)/dx^(n-1)+[/asciimath] [asciimath]...+a_1(x) (dy)/dx+a_0(x)y=f(x)[/asciimath]

where $[asciimath]y[/asciimath]$ is the dependent variable, $[asciimath]x[/asciimath]$ is the independent variable, $[asciimath]a\_i(x)[/asciimath]$are functions of $[asciimath]x[/asciimath]$(which can be constants or zeros), and $[asciimath]f(x)[/asciimath]$is a function of $[asciimath]x[/asciimath]$.

A nonlinear differential equation is one in which the dependent variable or its derivatives appear to a power greater than one, or they are multiplied together, or in any way that does not fit the linear form. For example, [asciimath]dy/dx+3y^2=3x[/asciimath] is nonlinear since [asciimath]y^2[/asciimath] has a power of 2.

E. Homogeneous and Nonhomogeneous Differential Equations

A differential equation is termed homogeneous if every term in the equation is a function of the dependent variable and its derivatives. For linear differential equations, an equation is homogeneous if the function [asciimath]f(x)[/asciimath] on the right-hand side of the equation is zero.

 [asciimath]a_n(x) (d^ny)/dx^n+a_(n-1)(x) (d^(n-1) y)/dx^(n-1)+[/asciimath] [asciimath]...+a_1(x) (dy)/dx+a_0(x)y=0[/asciimath]

For example, the linear equation [asciimath](d^2 y)/(dx^2 )+3 (dy)/(dx)+2y=0[/asciimath] is homogeneous because all terms are functions of $[asciimath]y[/asciimath]$and its derivatives, and the equation equals zero.

A differential equation is nonhomogeneous if it includes terms that are not solely functions of the dependent variable and its derivatives. For linear equations, this typically means there is a non-zero function on the right-hand side of the equation. For example, the linear equation [asciimath](d^2 y)/(dx^2 )+3 (dy)/(dx)+2y=xe^x[/asciimath] is nonhomogeneous because of the presence of the term [asciimath]xe^x[/asciimath], which is a function of the independent variable [asciimath]x[/asciimath] .

F. Solutions

A solution of a differential equation is a function that satisfies the equation on some open interval. This means that when the function and its derivatives are plugged into the differential equation, the equation holds true for all values within the interval. Often there are a set of solutions.

Now, consider the differential equation [asciimath]y' = 3x^2[/asciimath]. We can easily solve this equation by integrating:

[asciimath]dy/dx=3x^2[/asciimath]

[asciimath]dy=3x^2dx[/asciimath]

 [asciimath]int dy=int3x^2dx[/asciimath]

 [asciimath]y=x^3+C[/asciimath]

[asciimath]y=x^3+C[/asciimath], where [asciimath]C[/asciimath] is an arbitrary constant, represents a family of solutions to the given differential equation. Each distinct value of [asciimath]C[/asciimath] yields a unique particular solution, demonstrating how various initial conditions can be satisfied. This family of solutions, encompassing all possible solutions through the inclusion of the arbitrary constant [asciimath]C[/asciimath], is known as the general solution of the differential equation.

An explicit solution explicitly expresses the dependent variable in terms of the independent variable(s). For example, [asciimath]y=x^3+C[/asciimath] is an explicit solution. On the other hand, an implicit solution, may not directly express the dependent variable explicitly but still satisfies the differential equation. An example is [asciimath]y^2+x^2=C[/asciimath]. Note that finding an explicit solution is not always possible.

G. Initial Conditions

Initial condition(s) refer to the values specified for the dependent variable and possibly its derivatives at a specific point. Initial conditions are used to determine the specific (or particular) solution of a differential equation from the general solution, which typically contains arbitrary constants. For example, [asciimath]y(t_0)=y_0[/asciimath] states that at time  [asciimath]t_0[/asciimath], the value of $[asciimath]y[/asciimath]$is [asciimath]y_0[/asciimath]​. The number of initial conditions required for a given differential equation depends on the order of the differential equation. Generally, an [asciimath]nth[/asciimath] order differential equation needs [asciimath]n[/asciimath] initial conditions. These conditions specify the values of the function and its derivatives up to the [asciimath](n-1)th[/asciimath] order at a particular point. For example, a second-order differential equation requires two initial conditions. These are often the value of the function and the value of its first derivative at a specified point.

An Initial Value Problem (IVP) is a differential equation with initial condition(s) that nails down one particular solution. A solution might not be valid for all real numbers – there is the “interval of validity” or the domain of the solution.