Chapter 9 · Differential Equations

Exercise 9.1 · Q3: Order and degree of $\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0$

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.

3. $\left(\dfrac{ds}{dt}\right)^4 + 3s\dfrac{d^2s}{dt^2} = 0$

Exam answer

Here $s$ is the dependent variable and $t$ is the independent variable.

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

The equation is a polynomial equation in its derivatives $\dfrac{d^2s}{dt^2}$ and $\dfrac{ds}{dt}$ ⁠.

The highest power of $\dfrac{d^2s}{dt^2}$ 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 letters are new, the rules are not: $s$ is the dependent variable and $t$ the independent one. The highest derivative is $\frac{d^2s}{dt^2}$ ⁠, so the order is 2. The equation is a polynomial in $\frac{ds}{dt}$ and $\frac{d^2s}{dt^2}$ ⁠, so the degree is defined, and it is the power of the highest derivative $\frac{d^2s}{dt^2}$ ⁠, which is 1. The power 4 belongs to the lower derivative $\frac{ds}{dt}$ and does not count.

Given, to find, and the plan

Given: The differential equation $\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0$ ⁠, with $s$ the dependent variable and $t$ the independent variable.
To find: Its order, and its degree if the degree is defined.
Plan: (1) Identify the variables and list every derivative. (2) The order is the highest of them. (3) Test whether the equation is a polynomial equation in its derivatives. (4) Read off the power of the highest derivative only.

Basics you need

1Derivatives with other letters

$\frac{ds}{dt}$ is the first derivative of $s$ with respect to $t$ ⁠, and $\frac{d^2s}{dt^2}$ the second. The small 2 in $d^2s$ counts differentiations; it is not a power. 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

Identify the variables and list the derivatives

The derivatives are of $s$ with respect to $t$ ⁠, so $s$ is the dependent variable and $t$ the independent variable.

The derivatives present are $\dfrac{ds}{dt}$ (order 1) and $\dfrac{d^2s}{dt^2}$ (order 2). The $s$ in $3s$ is the variable itself, not a derivative.

The definitions of order and degree work for any letters: what matters is which quantity is being differentiated and with respect to what. Section 9.2 defines a differential equation through derivatives of the dependent variable with respect to the independent variable.

Differential equation and its variables, Section 9.2. Basics: y′ and y″
Read off the order

The highest order derivative present is $\dfrac{d^2s}{dt^2}$ ⁠. So
$\text{order} = 2.$
Section 9.2.1: the order is the order of the highest order derivative. The power 4 on $\frac{ds}{dt}$ does not make it a fourth derivative; it is still a first derivative, raised to the fourth power.

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

Write $p = \frac{ds}{dt}$ and $q = \frac{d^2s}{dt^2}$ ⁠. The equation reads
$p^4 + 3s\,q = 0.$
This is a polynomial in $p$ and $q$ ⁠: $p^4$ has a whole-number power, and $3s\,q$ is $q$ to the power 1 times the coefficient $3s$ ⁠. So the equation is a polynomial equation in its derivatives, and its degree is defined.

A coefficient that depends on the variables, such as $3s$ ⁠, is allowed: the condition only asks that the derivatives appear with whole-number powers and are not inside functions such as sin, cos, tan, log or e. The book's Example 1 (ii), $xy\frac{d^2y}{dx^2} + \ldots = 0$ ⁠, also has such a coefficient.

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

The highest order derivative is $q = \frac{d^2s}{dt^2}$ ⁠, and its highest power in the equation is 1. So
$\text{degree} = 1.$
The degree is the highest power of the highest order derivative only (Section 9.2.2). The largest power on the page, 4, is on $\frac{ds}{dt}$ ⁠, which is not the highest order derivative, so it is ignored.

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

Result

The equation $\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0$ has order 2 and degree 1. It is a polynomial equation in its derivatives, so the degree is defined.

(Textbook answer: Order 2; Degree 1.)

Check it with numbers

Hold $s = 0.5$ and $\frac{ds}{dt} = 1.2$ fixed. Then $(1.2)^4 = 2.0736$ and $3s = 1.5$ ⁠, so the left side is $2.0736 + 1.5\,\frac{d^2s}{dt^2}$ ⁠.

At $\frac{d^2s}{dt^2} = 0, 1, 2$ it is $2.0736$ ⁠, $3.5736$ ⁠, $5.0736$ ⁠: equal steps of 1.5. Equal steps mean $\frac{d^2s}{dt^2}$ enters to the first power, so the degree is 1.

Common mistakes

Answering "degree 4". The 4 is the power of $\frac{ds}{dt}$ ⁠, a lower derivative. Only the power of the highest order derivative counts.
Reading the small 2 in $\frac{d^2s}{dt^2}$ as a power. It counts differentiations; $\frac{d^2s}{dt^2}$ to the first power is still second order, first power.
Thinking the coefficient $3s$ spoils the polynomial. Coefficients may contain the variables; only derivatives inside functions such as sin, cos, tan, log or e spoil it.

What to take away

Change of letters changes nothing: find the dependent variable and read the derivatives of it.
The degree is the power of the highest order derivative, not the largest power you can see.

Every fact used, and where it comes from

Fact	Used in	Source
Dependent and independent variables in a differential equation	Step 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 coefficient such as $xy$ in front of $\frac{d^2y}{dx^2}$ is allowed	Step 3	Example 1 (ii), Section 9.2.2
Degree = highest power of the highest order derivative	Step 4	Section 9.2.2

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Exercise 9.1 · Q3: Order and degree of (dsdt)4+3sd2sdt2=0\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0(dtds​)4+3sdt2d2s​=0

Basics you need

Solution

Identify the variables and list the derivatives

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 · Q3: Order and degree of $\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0$