Classifying Differential Equations

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Section 5.1 Classifying Differential Equations

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Definition 5.2. Order of a DE.

The order of a differential equation is the order of the largest derivative that appears in the equation.

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Let's come back to our list of examples and state the order of each differential equation:

$y' = e^x\sec y$ has order 1
$y'-e^xy+3 = 0$ has order 1
$y'-e^xy = 0$ has order 1
$3y''-2y'=7$ has order 2
$4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5

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Definition 5.3. Linearity of a DE.

A linear differential equation can be written in the form

$\begin{equation*} F_n(x) \frac{d^ny}{dx^n}+F_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + F_2(x)\frac{d^2y}{dx^2} + F_1(x)\frac{dy}{dx} + F_0(x)y=G(x) \end{equation*}$

where $F_i(x)$ and $G(x)$ are functions of $x\text{.}$ Otherwise, we say that the differential equation is non-linear.

As an aside, if the leading coefficient

$F_n(x)$ is non-zero, then the equation is said to be of

$n$ -th order.

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Let's come back to our list of differential equations and add whether it is linear or not:

$y' = e^x\sec y$ has order 1, is non-linear
$y'-e^xy+3 = 0$ has order 1, is linear
$y'-e^xy = 0$ has order 1, is linear
$3y''-2y'=7$ has order 2, is linear
$4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear

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Definition 5.4. Homogeneity of a Linear DE.

Given a linear differential equation

$\begin{equation*} F_n(x) \frac{d^ny}{dx^n}+F_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + F_2(x)\frac{d^2y}{dx^2} + F_1(x)\frac{dy}{dx} + F_0(x)y=G(x) \end{equation*}$

where $F_i(x)$ and $G(x)$ are functions of $x\text{,}$ the differential equation is said to be homogeneous if $G(x)=0$ and non-homogeneous otherwise.

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Note: One implication of this definition is that

$y=0$ is a constant solution to a linear homogeneous differential equation, but not for the non-homogeneous case.

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Let's come back to all linear differential equations on our list and label each as homogeneous or non-homogeneous:

$y'-e^xy+3 = 0$ has order 1, is linear, is non-homogeneous
$y'-e^xy = 0$ has order 1, is linear, is homogeneous
$3y''-2y'=7$ has order 2, is linear, is non-homogeneous
$4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear, is homogeneous

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Definition 5.5. Linear DE with Constant Coefficients.

Given a linear differential equation

$\begin{equation*} F_n(x) \frac{d^ny}{dx^n}+F_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + F_2(x)\frac{d^2y}{dx^2} + F_1(x)\frac{dy}{dx} + F_0(x)y=G(x) \end{equation*}$

where $G(x)$ is a function of $x\text{,}$ the differential equation is said to have constant coefficients if $F_i(x)$ are constants for all $i\text{.}$

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As examples, we identify all linear differential equations on our list that have constant coefficients:

$y'-e^xy+3 = 0$ has order 1, is linear, is non-homogeneous, does not have constant coefficients
$y'-e^xy = 0$ has order 1, is linear, is homogeneous, does not have constant coefficients
$3y''-2y'=7$ has order 2, is linear, is non-homogeneous, has constant coefficients
$4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear, is homogeneous, does not have constant coefficients

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Example 5.6. Newton's Law of Cooling.

The equation from Newton's law of cooling,

$\begin{equation*} \frac{dy}{dt}=k(M-y) \end{equation*}$

is a first order linear non-homogeneous differential equation with constant coefficients, where $t$ is time, $k$ is the constant of proportionality, and $M$ is the ambient temperature.

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Exercises for Section 5.1.

Exercise 5.1.1.

Identify the order and linearity of each differential equation below.

$\ds{5y''' + 3y' - 4\sin(y) = \cos(x)}$
Answer
third order, non-linear
$\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$
Answer
third order, linear
$\ds{e^x\frac{dy}{dx}+e^{x+y} = e}$
Answer
first order, non-linear
$\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y^5 = 0}$
Answer
fourth order, non-linear
$\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$
Answer
second order, linear
$\ds{\tan(xy)y' = \sec(xy)}$
Answer
first order, non-linear

Exercise 5.1.2.

Identify the homogeneity of each linear differential equation below.

$\ds{5y''' + 3y' - 4y= \cos(x)}$
Answer
non-homogeneous
$\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$
Answer
homogeneous
$\ds{e^x\frac{dy}{dx}+e^{x}y = e}$
Answer
non-homogeneous
$\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y = 0}$
Answer
homogeneous
$\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$
Answer
non-homogeneous
$\ds{\tan(x)y' = \sec(x)y}$
Answer
homogeneous

Exercise 5.1.3.

State whether the coefficients of each linear differential equation below are constant or not.

$\ds{5y''' + 3y' - 4y= \cos(x)}$
Answer
The coefficients of the DE are 5,3 and 4. Therefore, the DE has constant coefficients.
$\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$
Answer
The coefficients are 4,2 and $-e^x\text{,}$ which means the DE has non-constant coefficients.
$\ds{e^x\frac{dy}{dx}+e^{x}y = e}$
Answer
The DE has non-constant coefficients, $e^x\text{.}$
$\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y = 0}$
Answer
The DE has non-constant coefficients, $x\text{,}$ $x^2$ and $x^3\text{.}$
$\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$
Answer
The DE has non-constant coefficients, $f_i(x)\text{.}$
$\ds{\tan(x)y' = \sec(x)y}$
Answer
The DE has non-constant coefficients, $\tan(x)$ and $\sec(x)\text{.}$