Introduction to Differential Equations

This set of notes serves as a brief refresher on differential equations and their applications in physics. Differential equations are equations that involve derivatives of a function, and they are used to describe how physical quantities change over time or space. They are ubiquitous in physics, especially in our study of quantum field theory, where they arise literally everywhere.

Table of Contents

• Table of Contents
• Classification of Differential Equations
• First-Order Differential Equations
  • Separable Equations
  • Linear First-Order Equations (Integrating Factor Method)
  • Exact First-Order Equations
  • Homogeneous Differential Equations
  • Bernoulli Equations

Classification of Differential Equations

A differential equation is an equation that relates a function to its derivatives. The solution to a differential equation is a function that satisfies the equation.

A general solution to a differential equation is a family of functions that contains all possible solutions to the equation. A particular solution is a specific function that satisfies the differential equation, often determined by initial or boundary conditions.

Differential equations can be classified in several ways, including by their order, linearity, and whether they are ordinary or partial differential equations.

• Ordinary Differential Equations (ODEs) involve derivatives with respect to a single independent variable, typically time. They are used to model systems that evolve over time, such as the motion of a particle or the growth of a population.

  Here's an example of an ODE, perhaps modeling a damped harmonic oscillator:

• Partial Differential Equations (PDEs) involve derivatives with respect to multiple independent variables, such as time and space. They are used to model systems that vary in both time and space, such as the propagation of waves or the diffusion of heat.

  An example of a PDE is the wave equation

  which describes the propagation of waves in a medium.

• Order refers to the highest derivative present in the equation. For example, the equation is a second-order ODE because it involves the second derivative of with respect to .

• Degree refers to the power of the highest derivative present in the equation. For example, the equation is a third-degree ODE because the highest derivative is raised to the third power.

• Linearity refers to whether the equation is linear or nonlinear. A linear differential equation is one in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. For example,

  is a linear first-order ODE, while

  is a nonlinear first-order ODE because of the term.

• A homogeneous differential equation is one in which all terms involve the dependent variable or its derivatives. For example,

  is a homogeneous first-order ODE, while

  is a non-homogeneous first-order ODE because of the term.

• An autonomous differential equation is one in which the independent variable does not explicitly appear in the equation. For example,

  is an autonomous first-order ODE, while

  is a non-autonomous first-order ODE because of the term.

Below is a table of some examples of differential equations, classified by their type, order, linearity, and whether they are homogeneous or autonomous.

Equation	Type	Order	Degree	Linearity	Homogeneous	Autonomous
	ODE	2	1	Linear	Yes	Yes
	PDE	2	1	Linear	Yes	Yes
	ODE	1	1	Linear	No	No
	ODE	1	1	Nonlinear	Yes	Yes
	ODE	1	1	Linear	Yes	Yes
	ODE	1	3	Linear	No	No
	ODE	1	1	Nonlinear	Yes	Yes
	PDE	1	1	Linear	Yes	No
	PDE	1	1	Linear	Yes	Yes

First-Order Differential Equations

Let's start with first-order differential equations, which involve the first derivative of a function. A general first-order ODE can be written in the form

where is the unknown function, and is a given function that describes how changes with respect to . We will look at five different cases of first-order ODEs: separable equations, linear equations, exact equations, homogeneous equations, and Bernoulli equations. The hardest part of solving a first-order ODE is usually recognizing which type of equation it is.

Separable Equations

If we can write as a product of two functions, one depending only on and the other only on , i.e.,

then we can separate the variables and write the equation as

(Is this step rigorously justified? Well, there are a few ways to go about this. One way is to use differential forms as we'll see when we study manifolds.) We can then integrate both sides to find the solution:

where is an arbitrary constant of integration.

Example: Consider the first-order ODE

We can separate the variables as follows:

Now, we can integrate both sides:

This is just

so

This is a general solution to the differential equation, where is an arbitrary constant.

Linear First-Order Equations (Integrating Factor Method)

Recall that a linear ODE is one in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. A general linear first-order ODE can be written in the form

where and are given functions. One convenient method to solve such equations is to multiply both sides by an integrating factor , which is defined as

Its derivative is

This is because allows us to rewrite the left-hand side of the equation as the derivative of a product, which simplifies the integration process. Multiplying both sides of the equation by , we get

As , we can rewrite the left-hand side as

We can then integrate both sides to yield

Example: Consider the first-order ODE

This is a non-separable equation, but it is linear. We can identify and . The integrating factor is

Multiplying both sides of the equation by , we get

The left-hand side can be rewritten to give

which means

Exact First-Order Equations

Consider a first-order ODE of the form

To derive the solution to such an equation, consider an equation of the form

where is a constant. The solution to this equation is a contour line of at .

Taking the total differential of both sides, we have

which is in the form of an exact first-order ODE with and .

The equation can be rearranged to give

where and . Under Schwarz's theorem, the mixed partial derivatives of are equal, so

This is the condition for the equation to be exact, meaning that there exists a function such that and . To solve an exact first-order ODE, we can integrate with respect to to find , and then set to find the solution.

Example: Consider the first-order ODE

We can identify and , so

We can check that the equation is exact by verifying that . The partial derivatives are

so the equation is indeed exact. To find , we just integrate with respect to :

where is an arbitrary function of . To determine , we can differentiate with respect to and set it equal to :

which implies that , so is a constant. Thus, we have

where is an arbitrary constant. The solution to the differential equation is then given by the contour lines of :

where is a constant.

Homogeneous Differential Equations

Previously, we defined a homogeneous differential equation as one in which all terms involve the dependent variable or its derivatives. However, there is another definition of homogeneity that is more general as it also applies to nonlinear differential equations.

A homogeneous function of degree is a function that satisfies the property

For example, the function is a homogeneous function of degree 2, since

Now, a homogeneous differential equation is a first-order ODE of the form

where and are homogeneous functions of the same degree. This is important because the quotient of two homogeneous functions of the same degree is a homogeneous function of degree 0. To show this, suppose and are homogeneous functions of degree . Then, by definition, we have

So

which shows that the quotient is a homogeneous function of degree 0. Let's label this function . Then, by the property of homogeneous functions of degree 0, we have

We can choose any scaling factor we want, so let's cleverly choose (assuming ). This gives

Plugging this back into our differential equation, we have

The term now only depends on the ratio . To solve this equation, we can define , where . The derivative of with respect to can be expressed in terms of as follows:

Thus, we can rewrite the equation as

This is a separable equation with the general solution

where is an arbitrary constant of integration. To proceed, we would need to know the specific form of .

Example: Consider the first-order ODE

We need to first check if the equation is homogeneous. We can identify and , which are both homogeneous functions of degree 2. Thus, the equation is indeed homogeneous.

We can then define . This is, as expected, a homogeneous function of degree 0. Making the substitution , we have

The differential equation can then be rewritten as

which simplifies to

This is a separable equation with the general solution

so

Bernoulli Equations

To reiterate what we have so far, a first-order ODE can be

• Separable:
• Linear:
• Exact: with

When a first-order ODE is homogeneous, we can make the substitution to convert it into a separable equation. When a first-order ODE is a Bernoulli equation, we can make the substitution to convert it into a linear equation.

A Bernoulli equation is a first-order ODE of the form

where and are given functions, and is a real number. To tackle such an equation, we need to tackle the problematic nonlinear term . First, we can divide both sides by (assuming ) to get

Now if we define , the second term just becomes , which looks perfectly linear. Differentiating with respect to , we have

Notice that this is just the first term of Equation multiplied by . Therefore, we can rewrite Equation as

so

This is a linear first-order ODE in whose coefficients are and . We can then solve this equation using the integrating factor method described earlier, in which the integrating factor is