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Section 2.3 Homogeneous equations with constant coefficients
Definition 2.3.1 . Spring/mass and circuit problems.
A second-order linear homogeneous ODE with constant coefficients takes the form
\begin{equation}
ay''+by'+cy=0,\tag{2.3.1}
\end{equation}
where \(a, b, c \in \mathbb{R}\) are constant functions.
Recall the first ODE we solved was of the form
\begin{equation}
ay'+by=0,\tag{2.3.2}
\end{equation}
which is a first-order linear homogeneous differential equation with constant coefficients. Recall that the solution was
\(y(t)=Ce^{rt}\) where
\(r=-\dfrac{b}{a}\text{.}\) Since
(2.3.1) and
(2.3.2) are of the same form, it is reasonable to expect that
\(y(t)=e^{rt}\) also solves
(2.3.1) on
\((-\infty,\infty)\text{.}\) However we need to know the value(s) of
\(r\) such that this is the case. Substituting
\(y(t)=e^{rt}\) into
(2.3.1) , we get
\begin{equation*}
ar^2e^{rt}+bre^{rt}+ce^{rt}=0.
\end{equation*}
We then factor out the common factor \(e^{rt}\) to get
\begin{equation*}
e^{rt}\big( ar^2+br+c \big)=0.
\end{equation*}
Since
\(e^{rt}\) is never zero, it can be divided off to leave a polynomial equation in
\(r\) that we can solve.
Definition 2.3.2 . Characteristic equation for (2.3.1).
The polynomial
\begin{equation*}
ar^2+br+c=0
\end{equation*}
is called the
characteristic equation (or
auxiliary equation ) for
(2.3.1) and its roots dictate the form of the general solution.
We now summarize what form general solutions take, dependent on the roots of the characteristic equation.
Table 2.3.3. General solutions to (2.3.1)
Two real, distinct roots \(r_1\neq r_2\)
\(y(t)=c_1e^{r_1t}+c_2e^{r_2t}\)
One real, repeated root \(r_1=-\dfrac{b}{2a}\)
\(y(t)=c_1e^{r_1t}+c_2te^{r_1t}\)
Complex roots \(r=\alpha \pm \beta i\)
\(y(t)=e^{\alpha t}\big[ c_1\cos(\beta t)+c_2\sin(\beta t)\big]\)
Example 2.3.4 . Real, distinct roots.
Consider the differential equation \(y''-5y'+6y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
Theorem 2.3.5 . Repeated roots.
Suppose
(2.3.1) has characteristic equation with repeated root
\(r_1=-\dfrac{b}{2a}\text{.}\) Then the general solution takes form
\begin{equation*}
y(t)=c_1e^{r_1t}+c_2te^{r_1t}.
\end{equation*}
Proof.
Example 2.3.6 . Real, repeated roots.
Consider the differential equation \(y''-4y'+4y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
Definition 2.3.7 . Euler’s formula.
Euler’s formula says
\begin{equation*}
e^{i\theta}=\cos(\theta)+i\sin(\theta).
\end{equation*}
Theorem 2.3.8 . Complex roots.
Suppose
(2.3.1) has a characteristic equation with complex roots
\(r=\alpha \pm \beta i\text{.}\) Then the general solution takes form
\begin{equation*}
y(t)=e^{\alpha t}\big[ c_1\cos(\beta t)+c_2\sin(\beta t) \big].
\end{equation*}
Proof.
Example 2.3.9 . Complex roots.
Consider the differential equation \(y''+4y'+13y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
Any linear homogeneous ODE with constant coefficients takes a form resembling
(2.3.2) and
(2.3.1) and we solve them the same way: assume a solution form
\(y(t)=e^{rt}\text{,}\) substitute it in, and solve the resulting polynomial for
\(r\text{.}\)
Example 2.3.10 .
Consider the differential equation \(y'''-2y''-y'+2y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
Example 2.3.11 .
Consider the differential equation \(y^{(4)}-16y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
Example 2.3.12 .
Consider the differential equation \(y^{(4)}-y'''-y'+y=0\text{.}\)
Classify the differential equation by order, linearity, type of coefficients, and whether or not the equation is homogeneous or nonhomogeneous.
Solve the differential equation.
