Differential Equations · Lesson 1 of 6

Recognising Differential Equations

“Learn to read equations involving derivatives and distinguish their order from their degree.”

Learning Objectives

• Recognise a differential equation and identify its dependent and independent variables. • Read prime and Leibniz notation for derivatives of different orders. • Determine the order of a differential equation. • Determine degree when the equation is polynomial in its derivatives, and explain when degree is undefined. • Separate the role of a derivative from its power and from functions of the variables.

Equations that describe change

Differentiation tells us how a known function changes. A differential equation starts with information about that change and asks which function could produce it. This is why differential equations can describe the growth of money or a population, as well as the slope of an unknown curve.

If $y = f (x)$ , the derivative $\frac{d y}{d x}$ measures the rate at which $y$ changes with $x$ . The equation $\frac{d y}{d x} = 2 x$ says that this rate equals twice the current value of $x$ . Integration suggests $y = x^{2} + C$ : changing $C$ changes the height of the curve but leaves its derivative unchanged. We will examine these families of functions in the next lesson.

Differential equations occur in physics, chemistry, biology, geology, economics, and other fields because many natural rules connect a quantity with its rate of change. A statement such as “the growth rate is proportional to the amount present” becomes an equation involving that amount and its derivative. The mathematical meaning of the derivative stays the same even when the variables represent different quantities.

Definition

Differential equation

An equation involving one or more derivatives of a dependent variable with respect to an independent variable, or independent variables, is a differential equation.

In $x \frac{d y}{d x} + y = 0$ , $x$ is the independent variable and $y$ depends on it. In $\frac{d s}{d t} = 3 t^{2}$ , $t$ is independent and $s$ is dependent. The letters are not what makes an equation differential: the presence of a derivative is the deciding feature.

Example — Recognise the defining feature

Problem
Which of $x^{2} - 3 x + 3 = 0$ , $sin x + cos x = 0$ , $x + y = 7$ , and $x \frac{d y}{d x} + y = 0$ is a differential equation?

1.The first three equations relate numbers or variable values without involving a derivative.
2.The fourth contains $\frac{d y}{d x}$ , the derivative of $y$ with respect to $x$ . Therefore it is a differential equation.
3.An equation can contain trigonometric functions and still fail to be differential; what matters here is differentiation, not the apparent complexity of the expression.

Definition

Ordinary differential equation

A differential equation involving derivatives with respect to only one independent variable is an ordinary differential equation. In this chapter, “differential equation” will mean an ordinary differential equation.

An equation involving derivatives with respect to more than one independent variable is called a partial differential equation. That distinction tells us the scope of our work; methods for partial differential equations are not needed here. For example, $2 \frac{d ^{2} y}{d x ^{2}} + (\frac{d y}{d x})^{3} = 0$ is ordinary even though it contains two different derivatives: both are taken with respect to $x$ .

Read derivative notation accurately

A derivative can be written in several equivalent ways. Recognising those forms prevents a notation change from looking like a new concept. Each successive derivative describes how the previous derivative changes.

Leibniz notation	Prime notation	Meaning
$\frac{d y}{d x}$	$y^{'}$	First derivative of y with respect to x
$\frac{d ^{2} y}{d x ^{2}}$	$y^{''}$	Second derivative: differentiate the first derivative
$\frac{d ^{3} y}{d x ^{3}}$	$y^{'''}$	Third derivative: differentiate the second derivative
$\frac{d ^{n} y}{d x ^{n}}$	$y^{(n)}$	Derivative of order n

The source also uses $y_{n}$ for a derivative of high order. We will normally use $y^{(n)}$ when many primes would be awkward. Neither $y^{(n)}$ nor the numerator $d^{n} y$ means the ordinary power $y^{n}$ . By contrast, $(\frac{d y}{d x})^{3}$ really is the cube of the first derivative.

Order of differentiation versus a power

$\frac{d ^{3} y}{d x ^{3}}$ is a third derivative, while $(\frac{d y}{d x})^{3}$ is a first derivative raised to the third power. Their superscripts have different jobs. Read the whole notation before classifying an equation.

Find the order first

Order records the highest level of differentiation that occurs anywhere in the equation. Ignore powers and coefficients at first, and look for the derivative taken the most times. This gives a useful first description of the equation even when its degree cannot be defined.

Definition

Order

The order of a differential equation is the order of its highest-order derivative.

Example — Three different orders

Problem
Find the order of $y^{'} = e^{x}$ , $y^{''} + y = 0$ , and $y^{'''} + x^{2} (y^{''})^{3} = 0$ .

1.The first equation contains only a first derivative, so its order is $1$ .
2.The second contains a second derivative, which is higher than any other derivative present, so its order is $2$ .
3.The third contains a third derivative and a cube of the second derivative. The highest derivative is still the third derivative, so the order is $3$ ; cubing the second derivative does not make it sixth order.

Example — Changing the variable letters

Problem
Find the order of $(\frac{d s}{d t})^{4} + 3 s \frac{d ^{2} s}{d t ^{2}} = 0$ .

1.The derivatives are of $s$ with respect to $t$ . There is a first derivative raised to the fourth power and a second derivative.
2.The highest derivative is $\frac{d ^{2} s}{d t ^{2}}$ , so the order is $2$ . The power $4$ is not the order.
3.The factor $s$ multiplying the second derivative does not change the order of the differential equation.

Decide whether degree is defined

Degree has an extra condition that order does not have. The equation must be polynomial in its derivatives. Think of the derivatives as the polynomial variables: their powers must be non-negative integers, and they cannot sit inside functions such as sine or the exponential.

The coefficients may depend on $x$ or $y$ . For example, $y^{''} + (y^{'})^{2} + sin y = 0$ is polynomial in the derivatives even though $sin y$ is not a polynomial in $y$ . In contrast, $y^{'} + sin (y^{'}) = 0$ contains the sine of a derivative and is not polynomial in the derivatives.

Definition

Degree

When a differential equation is polynomial in its derivatives, its degree is the highest power of its highest-order derivative. If the equation is not polynomial in its derivatives, degree is not defined under this definition.

Example — A high power of a lower derivative

Problem
Find order and degree of $y^{'''} + 2 (y^{''})^{2} - y^{'} + y = 0$ .

1.The highest derivative is $y^{'''}$ , so the order is $3$ .
2.Every derivative occurs polynomially. The highest-order derivative is raised to power $1$ , even though another derivative is squared.
3.Therefore the degree is $1$ , not $2$ . Degree is not the largest exponent that happens to appear anywhere.

Example — A nonlinear term without a derivative

Problem
Find order and degree of $(y^{'})^{2} + y^{'} - sin^{2} y = 0$ .

1.The only derivative is the first derivative, so the order is $1$ .
2.The equation is polynomial in $y^{'}$ . The term $sin^{2} y$ involves $y$ itself rather than a derivative, and is allowed as a coefficient or term.
3.The highest power of the highest-order derivative is $2$ , so the degree is $2$ .

Example — Classify three representative equations

Problem
Determine order and degree, if defined, for $y^{'} - cos x = 0$ ; $x y y^{''} + x (y^{'})^{2} - y y^{'} = 0$ ; and $y^{'''} + y^{2} + e^{y^{'}} = 0$ .

1.In the first equation the highest derivative is $y^{'}$ , occurring to power $1$ in a polynomial expression. Order $1$ , degree $1$ .
2.In the second equation the highest derivative is $y^{''}$ , so order is $2$ . Its power is $1$ , and all derivatives occur polynomially, so degree is $1$ .
3.In the third equation the third derivative makes the order $3$ . However, $e^{y^{'}}$ is an exponential function of a derivative. Degree is not defined.
4.The undefined degree in the final case does not prevent us from stating its order.

Example — A trigonometric function of a lower derivative

Problem
Determine order and degree of $\frac{d ^{4} y}{d x ^{4}} + sin (\frac{d ^{3} y}{d x ^{3}}) = 0$ .

1.The fourth derivative is present, so order is $4$ .
2.The sine of the third derivative prevents the equation from being polynomial in all its derivatives.
3.Degree is therefore undefined. Looking only at the power of the fourth derivative would miss the necessary polynomial check.

Check	What to inspect	Typical error
Order	Highest-order derivative present	Using the largest exponent
Degree exists?	Polynomial dependence on every derivative	Checking only the highest derivative
Degree value	Power of the highest-order derivative	Using the power of a lower derivative

Undefined is not zero

If a derivative occurs inside sine or an exponential, do not report degree zero. Say that degree is not defined. In the differential equations considered here, order and defined degree are positive integers.

Check your understanding

Use the same sequence each time: identify the highest derivative, test polynomial dependence on all derivatives, and only then state degree. The questions deliberately vary the variable letters and the position of nonlinear terms.

Quick check

What is the order of $(y^{'})^{7} + y^{''} = 0$ ?

Quick check

What is the degree of $(y^{''})^{3} + (y^{'})^{5} + y = 0$ ?

Quick check

Which term makes degree undefined in $y^{'''} + y^{2} + e^{y^{'}} = 0$ ?

Quick check

For $y^{''} + sin y = 0$ , which classification is correct?

Quick check

Which equation is an ordinary differential equation?

Quick check

What is the highest-order derivative in $y^{(4)} + (y^{''})^{6} = 0$ ?

Quick check

What is the degree of $\frac{d ^{4} y}{d x ^{4}} + sin (y^{'}) = 0$ ?

Quick check

In $\frac{d s}{d t} = s + t$ , which variable is independent?

Practice Problems

Determine order and degree, if defined: $y^{'} + 5 y = 0$ .
Determine order and degree: $(y^{'''})^{2} + (y^{''})^{3} + (y^{'})^{4} + y^{5} = 0$ .
Determine order and degree, if defined: $(y^{''})^{2} + cos (y^{'}) = 0$ .
Determine order and degree: $y^{''} = cos 3 x + sin 3 x$ .
Determine order and degree: $y^{'''} + 2 y^{''} + y^{'} = 0$ .
Determine order and degree: $y^{'} + y = e^{x}$ .
Determine order and degree: $y^{''} + (y^{'})^{2} + 2 y = 0$ .
Determine order and degree: $y^{''} + 2 y^{'} + sin y = 0$ .
Determine order and degree, if defined: $(y^{''})^{3} + (y^{'})^{2} + sin (y^{'}) + 1 = 0$ .
Determine order and degree: $2 x^{2} y^{''} - 3 y^{'} + y = 0$ .
Explain why the order of $(\frac{d s}{d t})^{4} + 3 s \frac{d ^{2} s}{d t ^{2}} = 0$ is not four.
Give one differential equation of order three whose degree is undefined, and explain the obstruction.

Order $1$ , degree $1$ .

Key Takeaways

• A differential equation relates a quantity to one or more of its derivatives. • Order is the order of the highest derivative, not the largest exponent. • Check polynomial dependence on every derivative before assigning degree. • When degree exists, it is the power of the highest-order derivative. • Functions of the variables themselves do not automatically make degree undefined.

Start of chapter

Next · Lesson 2

Solutions, Families, and Initial Conditions

Equations that describe change

Read derivative notation accurately

Find the order first

Decide whether degree is defined

Check your understanding

Quiz

Practice Problems

Key Takeaways