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Chapter 12 DE4: Homogeneous Second Order Linear DEs

So far we have learnt how to find exact solutions to separable first order DEs and linear first order DEs. For other types of first order DEs finding exact solutions can become quite involved (or even impossible) and so instead we will turn our attention to second order DEs. Recall that a second order DE is one that involves the second derivative of the unknown function. As with first order DEs we can only find analytic methods for solving second order DEs for certain classes of such DEs.

The class of second order DEs that we are going to consider in this lecture is called the class of “homogeneous, second order, linear DEs with constant coefficients”. Let's try to put this into some sort of context. Recall from Chapter 11 that a first order linear DE is a DE of the form:

\begin{equation*} P(x)y'+Q(x)y=f(x). \end{equation*}

As we have seen, this equation is called linear because it is the sum of terms that either don't involve \(y\) or only involve \(y\) or its derivatives raised to the power of one 1 . Extending this idea to second order DEs gives the following definition.

This is a very rudimentary statement of the concept of linearity

Definition 12.0.1.

A differential equation of the form,

\begin{equation*} P(x)y''+Q(x)y'+R(x)y=f(x) \text{.} \end{equation*}

where \(P(x)\neq 0 \) is called a second order linear DE.

In the case where \(f(x)= 0 \) the DE is called homogeneous. Thus:

Definition 12.0.2.

A differential equation of the form,

\begin{equation*} P(x)y''+Q(x)y'+R(x)y=0 \text{.} \end{equation*}

where \(P(x)\neq 0 \) is called a homogeneous second order linear DE.

Further, if each of \(P(x),\; Q(x), \text{and}\; R(x) \) and are constant functions then the DE is said to have constant coefficients. Thus:

Definition 12.0.3.

A differential equation of the form,

\begin{equation*} ay''+by'+cy=0 \text{.} \end{equation*}

where \(a,\; b, \; c \in \mathbb{R} \; \text{and}\; a\neq 0 \) is called a homogeneous second order linear DE with constant coefficients.