Calculus Early Transcendentals: Integral & Multi-Variable Calculus for Social Sciences

The 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. \(4\dfrac+ \cos x \dfrac= 0\) has order 5
Definition 5.3 . Linearity of a DE.

A differential equation can be written in the form

\begin F_n(x) \frac+F_(x)\frac + \dots + F_2(x)\frac + F_1(x)\frac + F_0(x)y=G(x) \end

where \(F_i(x)\) and \(G(x)\) are functions of \(x\text<.>\) Otherwise, we say that the differential equation is .

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. \(4\dfrac+ \cos x \dfrac= 0\) has order 5, is linear
Definition 5.4 . Homogeneity of a Linear DE.

Given a linear differential equation

\begin F_n(x) \frac+F_(x)\frac + \dots + F_2(x)\frac + F_1(x)\frac + F_0(x)y=G(x) \end

where \(F_i(x)\) and \(G(x)\) are functions of \(x\text\) the differential equation is said to be if \(G(x)=0\) and 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. \(4\dfrac+ \cos x \dfrac= 0\) has order 5, is linear, is homogeneous
Definition 5.5 . Linear DE with Constant Coefficients.

Given a linear differential equation

\begin F_n(x) \frac+F_(x)\frac + \dots + F_2(x)\frac + F_1(x)\frac + F_0(x)y=G(x) \end

where \(G(x)\) is a function of \(x\text\) the differential equation is said to have 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. \(4\dfrac+ \cos x \dfrac= 0\) 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 \frac=k(M-y) \end

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.
Exercise 5.1.1 .

Identify the order and linearity of each differential equation below.