Differential Equations · Study notes

MATOrder, Degree & Formation

Order & degree, and forming a differential equation

A differential equation relates a dependent variable $y$ to its derivatives with respect to an independent variable $x$ . The $\textbf{order}$ is the highest derivative present, and the $\textbf{degree}$ is the power of that highest-order derivative once the equation is made polynomial in all derivatives (free of radicals and fractional powers).

Forming a differential equation means starting from a family of curves with $n$ arbitrary constants and eliminating them by differentiating $n$ times, which is a guaranteed source of board marks.

General form of an order-2 equation

F\!\left(x,\,y,\,\dfrac{dy}{dx},\,\dfrac{d^2 y}{dx^2}\right)=0

F

is any relation; the order equals the highest derivative index appearing, here

2

Degree (after clearing radicals)

1+\left(\dfrac{dy}{dx}\right)^{2}=\left(k\,\dfrac{d^2 y}{dx^2}\right)^{2/3}\ \Rightarrow\ \left[1+\left(\dfrac{dy}{dx}\right)^{2}\right]^{3}=k^{2}\left(\dfrac{d^2 y}{dx^2}\right)^{2}

Degree

=2

(power of the highest-order derivative

\dfrac{d^2 y}{dx^2}

); degree is defined only after the equation is polynomial in derivatives.

Eliminating

n

constants

y=Ae^{2x}+Be^{-2x}\ \Rightarrow\ \dfrac{d^2 y}{dx^2}=4y

A,B

are the two arbitrary constants; two of them require differentiating twice, giving a second-order equation.

Family of lines through origin

y=mx\ \Rightarrow\ \dfrac{dy}{dx}=\dfrac{y}{x}

m

is the single arbitrary constant (slope); one constant gives a first-order, first-degree equation.

Circles touching

y

-axis at origin

x^{2}+y^{2}=2ax\ \Rightarrow\ \dfrac{dy}{dx}=\dfrac{y^{2}-x^{2}}{2xy}

a

is the arbitrary constant;

(x-a)^2+y^2=a^2

has centre

(a,0)

, radius

|a|

, and is tangent to the

y

-axis at the origin.

Order $=$ index of the highest derivative present; degree $=$ exponent of that highest derivative AFTER the equation is made a polynomial in all derivatives.
Before reading off the degree, remove all radicals and fractional/negative powers of derivatives by raising to a suitable power.
Degree is NOT defined if a derivative appears inside a transcendental function such as $\sin\!\left(\dfrac{dy}{dx}\right)$ , $e^{dy/dx}$ or $\log\!\left(\dfrac{dy}{dx}\right)$ .
To form a differential equation, count the arbitrary constants $n$ ; differentiate exactly $n$ times and eliminate the constants to get an $n$ -th order equation.
The resulting differential equation must be free of every arbitrary constant.
Family of straight lines through the origin $y=mx$ has $1$ constant $\Rightarrow$ order $1$ ; $y=Ae^{2x}+Be^{-2x}$ has $2$ constants $\Rightarrow$ order $2$ .
For $y^{2}=4ax$ (parabolas, vertex at origin, axis along $+x$ ): differentiate once to get $2y\dfrac{dy}{dx}=4a$ and eliminate $a$ to obtain $y=2x\dfrac{dy}{dx}$ .
Geometric families (circles, parabolas) are easiest if you first write the standard equation with the correct number of free constants, then differentiate.

Where the marks go

Reading off the degree before clearing the radical or fractional power — e.g. forgetting to cube both sides of $\left[1+\left(\dfrac{dy}{dx}\right)^{2}\right]=\left(k\,y''\right)^{2/3}$ first.
Claiming a degree exists when a derivative sits inside $\sin$ , $\cos$ , $\log$ or $e^{(\cdot)}$ — in those cases the degree is $\textbf{not defined}$ .
Differentiating the wrong number of times: too few leaves a constant behind, too many raises the order incorrectly.
Confusing order (which derivative) with degree (its power) — the highest-order derivative governs order regardless of the powers of lower derivatives.

How the board asks it

Numericalorder = highest derivative; degree = its power after clearing radicals and fractional powers
Find the order and degree (if defined) of the differential equation $\left[1+\left(\dfrac{dy}{dx}\right)^{2}\right]^{3/2}=k\,\dfrac{d^{2}y}{dx^{2}}$ .
Numericaleliminating $n$ arbitrary constants by differentiating $n$ times
Form the differential equation of the family of curves $y=Ae^{2x}+Be^{-2x}$ , where $A$ and $B$ are arbitrary constants.
Applicationgeometric family with one free constant: $x^{2}+y^{2}=2ax$
Obtain the differential equation of the family of circles touching the $y$ -axis at the origin.
Give reasonsdegree undefined when a derivative sits inside a transcendental function
State, giving reasons, whether the degree of $\dfrac{d^{2}y}{dx^{2}}+\sin\!\left(\dfrac{dy}{dx}\right)=0$ is defined.
Multiple choiceorder is the index of the highest derivative regardless of its power
The order and degree of $\left(\dfrac{d^{3}y}{dx^{3}}\right)^{2}+\left(\dfrac{dy}{dx}\right)^{4}=x$ are respectively $(a)\;3,2\quad(b)\;3,4\quad(c)\;2,3\quad(d)\;3,1$ .

Practise this topic

NextVariable separable equations

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.