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Section 5.1 Classifying Differential Equations

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.

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

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.

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

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.

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

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 \(i\text{.}\)

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

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.

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 \(\sec(x)\text{.}\)

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