Chapter 9 · Differential Equations

Exercise 9.1 · Q5: Order and degree of $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$

Exercise 9.1Order and degree · Sections 9.2.1 and 9.2.2

The question

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10.

5. $\dfrac{d^2y}{dx^2} = \cos 3x + \sin 3x$

Exam answer

The highest order derivative present in the equation is $\dfrac{d^2y}{dx^2}$ ⁠, so its order is 2.

The equation can be written as

\frac{d^2y}{dx^2} - \cos 3x - \sin 3x = 0.

The terms $\cos 3x$ and $\sin 3x$ contain no derivative, so this is a polynomial equation in $\dfrac{d^2y}{dx^2}$ ⁠.

The highest power of $\dfrac{d^2y}{dx^2}$ is 1, so its degree is 1.

$\therefore$ Order 2; degree 1.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The only derivative is $\frac{d^2y}{dx^2}$ ⁠, so the order is 2. The cosine and sine here are functions of $x$ ⁠, not of a derivative, so they do not stop the equation from being a polynomial in its derivatives. The degree is therefore defined, and it is the power of $\frac{d^2y}{dx^2}$ ⁠, which is 1.

Given, to find, and the plan

Given: The differential equation $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$ ⁠, with $y$ the dependent variable and $x$ the independent variable.
To find: Its order, and its degree if the degree is defined.
Plan: (1) List every derivative in the equation. (2) The order is the highest of them. (3) Bring everything to one side and test whether it is a polynomial equation in its derivatives. (4) Read off the power of the highest derivative.

Basics you need

1The second derivative

$\frac{d^2y}{dx^2} = y''$ is $y$ differentiated twice: how fast the slope $y'$ changes (Note 1, Section 9.2). Basics: y′ and y″

2Order

The order of a differential equation is the order of the highest order derivative in it (Section 9.2.1). Lesson: order and degree

3Degree, and its condition

The degree is defined only when the equation is a polynomial equation in its derivatives. Then it is the highest power (a positive integer) of the highest order derivative (Section 9.2.2). Lesson: order and degree

Solution

List the derivatives in the equation

The only derivative is $\dfrac{d^2y}{dx^2}$ ⁠, of order 2. The right side, $\cos 3x + \sin 3x$ ⁠, is a function of $x$ alone and contains no derivative.

Order and degree are about derivatives of the dependent variable $y$ ⁠. The independent variable $x$ ⁠, and functions of $x$ alone, are not derivatives.

Notation $y'' = \frac{d^2y}{dx^2}$ ⁠: Note 1, Section 9.2. Basics: y′ and y″
Read off the order

The highest order derivative present is $\dfrac{d^2y}{dx^2}$ ⁠. So
$\text{order} = 2.$
Section 9.2.1: the order is the order of the highest order derivative. It does not matter that there is no $y'$ or $y$ in the equation.

Definition of order, Section 9.2.1. Lesson: order and degree
Is it a polynomial equation in its derivatives?

Bring every term to the left and write $q = \frac{d^2y}{dx^2}$ ⁠:
$q - \cos 3x - \sin 3x = 0.$
As a polynomial in $q$ ⁠, this has the term $q$ (power 1) and the term $-\cos 3x - \sin 3x$ ⁠, which has no $q$ in it (power 0). So the equation is a polynomial equation in its derivatives, and its degree is defined.

A sine or cosine breaks the polynomial condition only when a derivative sits inside it. Here the cosine and sine act on $3x$ ⁠, and $3x$ is not a derivative. The book makes the same point with equation (10) in Section 9.2.2: a polynomial in $y'$ even though it is not a polynomial in $y$ ⁠.

Condition for the degree, Section 9.2.2. Lesson: Example 1, order and degree
Read off the degree

The highest order derivative is $\frac{d^2y}{dx^2}$ ⁠, and its highest power in the equation is 1. So
$\text{degree} = 1.$
By Section 9.2.2, the degree of a polynomial equation in the derivatives is the highest power of the highest order derivative.

Definition of degree, Section 9.2.2. Lesson: Exercise 9.1 player

Result

The equation $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$ has order 2 and degree 1. The sine and cosine are of $x$ ⁠, not of a derivative, so it is a polynomial equation in its derivatives.

(Textbook answer: Order 2; Degree 1.)

Check it with numbers

Take $x = 0.2$ ⁠, so $3x = 0.6$ ⁠: $\cos 0.6 + \sin 0.6 \approx 1.389978$ ⁠. The left side minus the right side is $\frac{d^2y}{dx^2} - 1.389978$ ⁠. At $\frac{d^2y}{dx^2} = 0, 1, 2$ it is $-1.389978$ ⁠, $-0.389978$ ⁠, $0.610022$ ⁠: equal steps of 1, so $\frac{d^2y}{dx^2}$ enters to the first power.

The equation is also easy to satisfy: $y = -\frac19(\cos 3x + \sin 3x)$ works, because differentiating twice multiplies $\cos 3x$ and $\sin 3x$ by $-9$ ⁠. At $x = 0.2$ its second derivative is $\cos 0.6 + \sin 0.6 \approx 1.389978$ ⁠, the right side.

Common mistakes

Saying "degree not defined" because of cos and sin. They act on $3x$ ⁠, not on a derivative, so the equation is still a polynomial in its derivatives.
Reading the 2 in $\frac{d^2y}{dx^2}$ as the degree. It is the order (two differentiations); the power is 1.
Reading $\cos 3x$ as $\cos^3 x$ ⁠. The book prints $\cos 3x + \sin 3x$ ⁠; either way the answer would not change, since neither contains a derivative.

What to take away

Functions of $x$ (or of $y$ ⁠) are harmless for the degree; only functions of a derivative spoil it.
An equation $\frac{d^2y}{dx^2} = f(x)$ is second order, first degree.

Every fact used, and where it comes from

Fact	Used in	Source
$y'' = \frac{d^2y}{dx^2}$	Step 1	Note 1, Section 9.2
Order = order of the highest order derivative	Step 2	Section 9.2.1 (Lesson: order and degree)
Degree is defined only for a polynomial equation in the derivatives	Step 3	Section 9.2.2
A polynomial in $y'$ need not be a polynomial in $y$	Step 3	Equation (10), Section 9.2.2
Degree = highest power of the highest order derivative	Step 4	Section 9.2.2
$\frac{d}{dx}\cos 3x = -3\sin 3x$ ⁠, $\frac{d}{dx}\sin 3x = 3\cos 3x$	Check it with numbers	Chain rule, Chapter 5

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Exercise 9.1 · Q5: Order and degree of d2ydx2=cos⁡3x+sin⁡3x\frac{d^2y}{dx^2} = \cos 3x + \sin 3xdx2d2y​=cos3x+sin3x

Basics you need

Solution

List the derivatives in the equation

Read off the order

Is it a polynomial equation in its derivatives?

Read off the degree

Every fact used, and where it comes from

Exercise 9.1 · Q5: Order and degree of $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$