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## Section5.1Classifying Differential Equations

###### Definition5.2. Order of a DE.

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

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

1. $y' = e^x\sec y$ has order 1

2. $y'-e^xy+3 = 0$ has order 1

3. $y'-e^xy = 0$ has order 1

4. $3y''-2y'=7$ has order 2

5. $4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5

###### Definition5.3. 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.

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

1. $y' = e^x\sec y$ has order 1, is non-linear

2. $y'-e^xy+3 = 0$ has order 1, is linear

3. $y'-e^xy = 0$ has order 1, is linear

4. $3y''-2y'=7$ has order 2, is linear

5. $4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear

###### Definition5.4. 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.

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.

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

1. $y'-e^xy+3 = 0$ has order 1, is linear, is non-homogeneous

2. $y'-e^xy = 0$ has order 1, is linear, is homogeneous

3. $3y''-2y'=7$ has order 2, is linear, is non-homogeneous

4. $4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear, is homogeneous

###### Definition5.5. 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 $i\text{.}$

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

1. $y'-e^xy+3 = 0$ has order 1, is linear, is non-homogeneous, does not have constant coefficients

2. $y'-e^xy = 0$ has order 1, is linear, is homogeneous, does not have constant coefficients

3. $3y''-2y'=7$ has order 2, is linear, is non-homogeneous, has constant coefficients

4. $4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0$ has order 5, is linear, is homogeneous, does not have constant coefficients

###### Example5.6. 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.

##### Exercises for Section 5.1.

Identify the order and linearity of each differential equation below.

1. $\ds{5y''' + 3y' - 4\sin(y) = \cos(x)}$

Answer
third order, non-linear
2. $\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$

Answer
third order, linear
3. $\ds{e^x\frac{dy}{dx}+e^{x+y} = e}$

Answer
first order, non-linear
4. $\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y^5 = 0}$

Answer
fourth order, non-linear
5. $\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$

Answer
second order, linear
6. $\ds{\tan(xy)y' = \sec(xy)}$

Answer
first order, non-linear

Identify the homogeneity of each linear differential equation below.

1. $\ds{5y''' + 3y' - 4y= \cos(x)}$

Answer
non-homogeneous
2. $\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$

Answer
homogeneous
3. $\ds{e^x\frac{dy}{dx}+e^{x}y = e}$

Answer
non-homogeneous
4. $\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y = 0}$

Answer
homogeneous
5. $\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$

Answer
non-homogeneous
6. $\ds{\tan(x)y' = \sec(x)y}$

Answer
homogeneous

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

1. $\ds{5y''' + 3y' - 4y= \cos(x)}$

Answer

The coefficients of the DE are 5,3 and 4. Therefore, the DE has constant coefficients.

2. $\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}$

Answer

The coefficients are 4,2 and $-e^x\text{,}$ which means the DE has non-constant coefficients.

3. $\ds{e^x\frac{dy}{dx}+e^{x}y = e}$

Answer

The DE has non-constant coefficients, $e^x\text{.}$

4. $\ds{-\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}-x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}-x^4y = 0}$

Answer

The DE has non-constant coefficients, $x\text{,}$ $x^2$ and $x^3\text{.}$

5. $\ds{f_2(x)y''+f_1(x)y'+f_0(x)y-1=0}\text{,}$ where $f_i(x)$ are non-constant functions of $x$

Answer

The DE has non-constant coefficients, $f_i(x)\text{.}$

6. $\ds{\tan(x)y' = \sec(x)y}$

Answer

The DE has non-constant coefficients, $\tan(x)$ and $\sec(x)\text{.}$