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:
\(y' = e^x\sec y\) 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
Definition 5.3. Linearity of a DE.
A linear differential equation can be written in the form
where \(F_i(x)\) and \(G(x)\) are functions of \(x\text{.}\) Otherwise, we say that the differential equation is nonlinear.
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 nonlinear
\(y'e^xy+3 = 0\) 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
Definition 5.4. Homogeneity of a Linear DE.
Given a linear differential 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 nonhomogeneous 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 nonhomogeneous case.
Let's come back to all linear differential equations on our list and label each as homogeneous or nonhomogeneous:
\(y'e^xy+3 = 0\) has order 1, is linear, is nonhomogeneous
\(y'e^xy = 0\) has order 1, is linear, is homogeneous
\(3y''2y'=7\) has order 2, is linear, is nonhomogeneous
\(4\dfrac{d^5y}{dx^5} + \cos x \dfrac{dy}{dx} = 0\) has order 5, is linear, is homogeneous
Definition 5.5. Linear DE with Constant Coefficients.
Given a linear differential equation
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:
\(y'e^xy+3 = 0\) has order 1, is linear, is nonhomogeneous, 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 nonhomogeneous, 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
Example 5.6. Newton's Law of Cooling.
The equation from Newton's law of cooling,
is a first order linear nonhomogeneous 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.
Exercise 5.1.1.
Identify the order and linearity of each differential equation below.

\(\ds{5y''' + 3y'  4\sin(y) = \cos(x)}\)
Answerthird order, nonlinear 
\(\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}\)
Answerthird order, linear 
\(\ds{e^x\frac{dy}{dx}+e^{x+y} = e}\)
Answerfirst order, nonlinear 
\(\ds{\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}x^4y^5 = 0}\)
Answerfourth order, nonlinear 
\(\ds{f_2(x)y''+f_1(x)y'+f_0(x)y1=0}\text{,}\) where \(f_i(x)\) are nonconstant functions of \(x\)
Answersecond order, linear 
\(\ds{\tan(xy)y' = \sec(xy)}\)
Answerfirst order, nonlinear
Exercise 5.1.2.
Identify the homogeneity of each linear differential equation below.

\(\ds{5y''' + 3y'  4y= \cos(x)}\)
Answernonhomogeneous 
\(\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}\)
Answerhomogeneous 
\(\ds{e^x\frac{dy}{dx}+e^{x}y = e}\)
Answernonhomogeneous 
\(\ds{\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}x^4y = 0}\)
Answerhomogeneous 
\(\ds{f_2(x)y''+f_1(x)y'+f_0(x)y1=0}\text{,}\) where \(f_i(x)\) are nonconstant functions of \(x\)
Answernonhomogeneous 
\(\ds{\tan(x)y' = \sec(x)y}\)
Answerhomogeneous
Exercise 5.1.3.
State whether the coefficients of each linear differential equation below are constant or not.

\(\ds{5y''' + 3y'  4y= \cos(x)}\)
AnswerThe coefficients of the DE are 5,3 and 4. Therefore, the DE has constant coefficients.

\(\ds{4\frac{d^3y}{dx^3}+2\frac{dy}{dx} = e^x y}\)
AnswerThe coefficients are 4,2 and \(e^x\text{,}\) which means the DE has nonconstant coefficients.

\(\ds{e^x\frac{dy}{dx}+e^{x}y = e}\)
AnswerThe DE has nonconstant coefficients, \(e^x\text{.}\)

\(\ds{\frac{d^4y}{dx^4}+x\frac{d^3y}{dx^3}x^2\frac{d^2y}{dx^2}+x^3\frac{dy}{dx}x^4y = 0}\)
AnswerThe DE has nonconstant coefficients, \(x\text{,}\) \(x^2\) and \(x^3\text{.}\)

\(\ds{f_2(x)y''+f_1(x)y'+f_0(x)y1=0}\text{,}\) where \(f_i(x)\) are nonconstant functions of \(x\)
AnswerThe DE has nonconstant coefficients, \(f_i(x)\text{.}\)

\(\ds{\tan(x)y' = \sec(x)y}\)
AnswerThe DE has nonconstant coefficients, \(\tan(x)\) and \(\sec(x)\text{.}\)