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Handout Lessons 15 and 16, Homogeneous Equations with Constant Coefficients
This lesson is based on Section 3.3 of your textbook by Edwards, Penney, and Calvis.
Recall that an \(n\) th-order homogeneous linear differential equation
\begin{equation*}
p_0(x)y^{(n)}+p_1(x)y^{(n-1)}+\dots+p_{n-1}(x)y'+p_n(x)y=0
\end{equation*}
Today we are going to consider the special case in which the coefficient functions are all constants.
Definition 110 .
An \(n\) th-order homogeneous linear differential equation with constant coefficients
\begin{equation*}
a_ny^{(n)}+a_{n-1}y^{(n-1)}+\dots+a_{1}y'+a_0y=0
\end{equation*}
where \(a_0, a_1, \dots, a_{n-1}, a_n\) are all constants.
We start with some notation that will simplify the writing of derivatives.
Definition 111 .
\(D^n\) is a differential operator . It acts on a function by the rule
\begin{equation*}
D^ny=y^{(n)}
\end{equation*}
In particular,
\(Dy=\)
\(D^2y=\)
\(D^6y=\)
You can take linear combinations of differential operators, so that
\begin{equation*}
(2D^3+5D^2-3D+4)y=\fillinmath{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX}
\end{equation*}
The following lemma shows that differential operators commute.
Lemma 112 .
For constants \(a\) and \(b\text{,}\)
\begin{equation*}
(D-a)(D-b)=(D-b)(D-a)
\end{equation*}
Real Roots of the Characteristic Equation.
If we start with an \(n\) th-order homogeneous linear differential equation with constant coefficients
\begin{gather}
a_ny^{(n)}+a_{n-1}y^{(n-1)}+\dots+a_{1}y'+a_0y=0\tag{✶}
\end{gather}
we can rewrite this equation using operator notation:
\begin{equation*}
\fillinmath{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX}
\end{equation*}
If we replace the
\(D\) ’s in the operator with the numerical variable
\(r\text{,}\) we obtain the characteristic equation of
(✶) .
Definition 113 .
The characteristic equation of the \(n\) th-order homogeneous linear differential equation with constant coefficients
\begin{equation*}
a_ny^{(n)}+a_{n-1}y^{(n-1)}+\dots+a_{1}y'+a_ny=0
\end{equation*}
is
\begin{equation*}
a_nr^n+a_{n-1}r^{n-1}+\dots+a_{1}r+a_0=0
\end{equation*}
Because the
\(r\) ’s are numerical variables, we can apply Algebra to the characteristic equation. In particular, we can factor the characteristic equation. It turns out that the roots of the characteristic equation allow us to identify the components of the general solution of
(✶) .
\(\alpha\text{,}\) real, multiplicity \(1\)
\(Ce^{\alpha x}\)
\(\alpha\text{,}\) real, multiplicity \(k\)
\((C_1+C_2x+C_3x^2+\dots+C_kx^{k-1})e^{\alpha x}\)
Example 114 . Using the characteristic equation.
(Number 6 from Section 3.3 of your textbook by Edwards, et.al.)
Find the general solution of the differential equation.
\begin{equation*}
y''+5y'+5y=0\text{.}
\end{equation*}
Example 115 . Using the characteristic equation.
(Number 10 from Section 3.3 of your textbook by Edwards, et.al.)
Find the general solution of the differential equation.
\begin{equation*}
5y^{(4)}+3y^{(3)}=0\text{.}
\end{equation*}
The following theorem from Algebra may be helpful in solving the next problem.
Theorem 116 . Rational roots theorem.
Consider the polynomial \(a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\) with integer coefficients and \(a_n \neq 0\text{.}\) If it has any rational roots, then the roots are of the form:
\begin{equation*}
\pm \frac{p}{t}
\end{equation*}
where \(p\) is a factor of \(a_0\) and \(t\) is a factor of \(a_n\text{.}\)
Example 117 . Using the characteristic equation.
(Number 32 from Section 3.3 of your textbook by Edwards, et.al.)
Find the general solution of the differential equation.
\begin{equation*}
y^{(4)}+y^{(3)}-3y''-5y'-2y=0\text{.}
\end{equation*}
Example 118 .
The general solution of a homogeneous differential equation with constant coefficients is
\begin{equation*}
y(x)=(c_1+c_2x)e^{2x}+c_3e^{-3x}\text{.}
\end{equation*}
What is the differential equation?
We now turn our attention to complex roots of the characteristic equation. In order to do this, we need to develop an understanding of
\begin{equation*}
e^{ri}
\end{equation*}
where \(r\) is a real number and \(i\) is the imaginary number that satisfies
\begin{align*}
i^2 = & \fillinmath{XXXXXXXXXX}\\
i^3 = & \fillinmath{XXXXXXXXXX}\\
i^4 = & \fillinmath{XXXXXXXXXX}\\
i^5 = & \fillinmath{XXXXXXXXXX}\\
\end{align*}
For reasons that should be evident very soon, it is typical use
\(\theta\) instead of
\(r\text{.}\) From our knowledge of power series, we then have that
Theorem 119 . Euler’s Formula for Complex Powers of \(e\) .
For the real number \(\theta\) and the complex number \(i\) (\(i^2=-1\) ),
\begin{equation*}
e^{i\theta}=\cos(\theta)+i\sin(\theta)
\end{equation*}
Complex Roots of the Characteristic Equation.
We now wish to understand what complex roots of the characteristic equation
\begin{gather}
a_nr^n+a_{n-1}r^{n-1}+\dots+a_{1}r+a_0=0\tag{#}
\end{gather}
tell us about the general solution of
\begin{gather}
a_ny^{(n)}+a_{n-1}y^{(n-1)}+\dots+a_{1}y'+a_0y=0\tag{†}
\end{gather}
Complex roots of a polynomial equation come in pairs
\begin{equation*}
a\pm bi \qquad a, b \in \mathbb{R}
\end{equation*}
If this pair of roots has multiplicity 1 and follows the same pattern as real roots of multiplicity 1, then what should they contribute to the general solution of
(†) ?
\(\alpha\text{,}\) real, multiplicity \(1\)
\(Ce^{\alpha x}\)
\(\alpha\text{,}\) real, multiplicity \(k\)
\((C_1+C_2x+C_3x^2+\dots+C_kx^{k-1})e^{\alpha x}\)
\(a \pm bi\text{,}\) complex pair, multiplicity \(1\)
\(e^{ax}(C_1\cos(bx)+C_2\sin(bx))\)
\(a \pm bi\text{,}\) complex pair, multiplicity \(k\)
\(\sum_{i=0}^{k-1}x^ie^{ax}(C_i\cos(bx)+D_i\sin(bx))\)
Example 120 . Complex roots of the characteristic equation.
(Number 16 from Section 3.3 of your textbook by Edwards, et.al.)
Find the general solution of the differential equation.
\begin{equation*}
y^{(4)}+18y''+81y=0\text{.}
\end{equation*}
Example 121 . Complex roots of the characteristic equation.
(Number 14 from Section 3.3 of your textbook by Edwards, et.al.)
Find the general solution of the differential equation.
\begin{equation*}
y^{(4)}+3y''-4y=0\text{.}
\end{equation*}
Example 122 . Complex roots of the characteristic equation.
(Number 36 from Section 3.3 of your textbook by Edwards, et.al.)
You are told that \(y=e^{-x}\sin(x)\) is a solution of
\begin{equation*}
9y^{(3)}+11y''+4y'-14y=0
\end{equation*}
Find the general solution of this differential equation.
Example 123 . A substitution problem.
(Number 53 from Section 3.3 of your textbook by Edwards, et.al.)
Use the substitution \(v=\ln(x)\) to solve
\begin{equation*}
x^2y''+7xy'+25y=0
\end{equation*}
