Definition 75.
A second-order differential equation in the function \(y(x)\) is a differential equation equivalent to an equation of the form:
\begin{equation*}
F(x,y,y',y'')=0\text{.}
\end{equation*}
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Handout Lesson 13, Second-Order Linear Differential Equations
Textbook Section(s).
This lesson is based on Section 3.1 of your textbook by Edwards, Penney, and Calvis.
Definitions.
Definition 76.
A second-order differential equation is linear if it is equivalent to an equation of the form
\begin{equation*}
A(x)y''+B(x)y'+C(x)y=D(x)
\end{equation*}
(In order to be a second-order differential equation, we require \(A(x)\) is not the zero function.)
Unless otherwise noted, you should assume that \(A(x)\text{,}\) \(B(x)\text{,}\) \(C(x)\text{,}\) and \(D(x)\) are continuous on an open interval \(I\text{.}\)
Example 77.
Which of the following are linear differential equations?
-
\(\displaystyle y''+2xy'+e^xy=x^3\)
-
\(\displaystyle (2y+1)y''+(x-3)y'+xy=x^3\)
-
\(\displaystyle y''+2(y')^2+3y=5\)
Definition 78.
A second-order linear differential equation is homogenous if it is equivalent to an equation of the form
\begin{gather}
A(x)y''+B(x)y'+C(x)y=0.\tag{✶}
\end{gather}
Second-order linear differential equations that are not homogenous are called nonhomogenous.
Recall that we have assumed that \(A(x)\text{,}\) \(B(x)\text{,}\) and \(C(x)\) are continuous on an open interval \(I\text{.}\) If we also assume that \(A(x)\neq 0\) for each \(x \in I\text{,}\) then we may divide (✶) by \(A(x)\) and write our equation as
\begin{gather}
y''+p(x)y'+q(x)y=0\tag{†}
\end{gather}
Every nonhomogenous linear equation has an associated homogenous linear equation.
Example 79.
Find the associated homogenous linear equation of the nonhomogenous linear equation
\begin{equation*}
x^3y''-\cos(x)=e^xy'+5xy
\end{equation*}
Application: In section 3.1, your textbook authors show you that the vibrations of a mass that is attached to both a spring and a dashpot can be modeled by a second-order linear differential equation. I encourage you to read this example because we will come back to it in a few sections.
Solutions of homogenous linear differential equations.
Theorem 80. Principle of Superposition for Homogenous Equations.
If \(y_1\) and \(y_2\) are solutions of the homogenous linear differential equation
\begin{gather}
A(x)y''+ B(x)y'+C(x)y=0\tag{#}
\end{gather}
and \(c_1, c_2\) are constants, then
\begin{equation*}
y=c_1y_1+c_2y_2
\end{equation*}
is also a solution to (#).
Existence and uniqueness of solutions for linear differential equations.
Theorem 82. Existence and Uniqueness of Solutions for Second-Order Linear Differential Equations.
Assume that \(p\text{,}\) \(q\text{,}\) and \(f\) are continuous on the open interval \(I\) that contains the point \(x=a\text{.}\) Then for any two constants \(b_0\) and \(b_1\text{,}\) there is a unique solution on \(I\) for the initial value problem
\begin{gather*}
y''+p(x)y'+q(x)u = f(x) \\
y(a)=b_0\\
y'(a)=b_1
\end{gather*}
Example 83. Solving an IVP with a linear differential equation.
(Exercise 4 in section 3.1 of your textbook)
\begin{gather}
y''+25y=0\tag{††}
\end{gather}
Then find constants \(c_1\) and \(c_2\) so that \(y=c_1y_1+c_2y_2\) is a solution to (††) that satisfies the initial conditions \(y(0)=10\) and \(y'(0)=-10\text{.}\)
Linear independence and general solutions of linear differential equations.
Definition 84. Linear Independence of Two Functions.
Two functions \(f\) and \(g\) are linearly independent on an open interval \(I\) provided that neither function is a scalar multiple of the other on \(I\text{.}\)
If there is no constant \(k\) such that
\begin{equation*}
\frac{f(x)}{g(x)}=k \text{ or } \frac{g(x)}{f(x)}=k \text{ for all } x\in
I\text{,}
\end{equation*}
then \(f\) and \(g\) are linearly independent on \(I\text{.}\)
Example 85.
Consider each pair of functions that are continuous on \((-\infty,\infty)\text{.}\) Which pairs are linearly independent?
-
\(e^x\) and \(e^{2x}\)
-
\(e^x\) and \(e^{x+2}\)
-
\(|x|\) and \(x\)
-
\(\sin(x)\) and \(\cos(x)\)
-
\(0\) and \(x^2\)
Definition 86. Wronskian.
The Wronskian, \(W(f,g)\text{,}\) of the functions \(f\) and \(g\) is defined by
\begin{align}
W(f,g)=\left\lvert
\begin{array}{cc}
f(x) & g(x) \\
f'(x) & g'(x) \\
\end{array}
\right\rvert\tag{##}
\end{align}
Note: \(W(f,g)\) is used to denote a function of \(x\text{,}\) so you may also see the Wronskian written as \(W(x)\) if we wish to emphasize that it can be evaluated at \(x\text{.}\)
Theorem 88. Wronskians and Solutions of Linear Differenital Equations.
Assume \(p(x)\) and \(q(x)\) are continuous on the open interval \(I\) and that \(y_1\) and \(y_2\) are solutions to the homogenous equation
\begin{gather*}
y''+p(x)y'+q(x)y=0
\end{gather*}
-
\(y_1\) and \(y_2\) are linearly independent \(\implies\) \(W(y_1,y_2) \neq 0\) at each point \(x \in I\text{.}\)
Theorem 89.
(Part of the discussion in your book, but not listed as a theorem )
If \(p(x)\) and \(q(x)\) are continuous on the open interval \(I\text{,}\) then the homogenous equation
\begin{equation*}
y''+p(x)y'+q(x)y=0
\end{equation*}
always has two linearly independent solutions on \(I\text{.}\)
Theorem 90. General Solutions of Homogenous Linear Differential Equations.
Assume \(p(x)\) and \(q(x)\) are continuous on the open interval \(I\) and that \(y_1\) and \(y_2\) are linearly independent solutions to the homogenous equation
\begin{gather}
y''+p(x)y'+q(x)y=0\tag{✶✶✶}
\end{gather}
on \(I\text{.}\) If \(Y\) is any solution of (✶✶✶) on \(I\text{,}\) then there are constants \(c_1\) and \(c_2\) such that
\begin{equation*}
Y(x)=c_1y_1(x)+c_2y_2(x)
\end{equation*}
on \(I\text{.}\)
Example 91. Linear combinations of solutions of linear differential equations.
In example 4 of section 3.1 of your textbook, your textbook authors show you that the homogenous linear equation
\begin{equation*}
y''-4y=0
\end{equation*}
has solutions
\begin{equation*}
y_1=e^{2x}, \qquad y_2=e^{-2x}, \text{ and} \qquad y_3=\cosh(2x)
\end{equation*}
Since \(y_1\) and \(y_2\) are linearly independent, the previous theorem implies that \(\cosh(2x)\) must be a linear combination of \(y_1\) and \(y_2\text{.}\) In fact, you already knew this to be true.
Linear second-order differential equations with constant coefficients.
In this section we consider the linear second-order differential equation
\begin{gather*}
ay''+by'+cy=0
\end{gather*}
where \(a\text{,}\) \(b\text{,}\) and \(c\) are constants.
Definition 92.
The characteristic equation of
\begin{equation*}
ay''+by'+cy=0
\end{equation*}
is
\begin{equation*}
ar^2+br+c=0\text{.}
\end{equation*}
Theorem 93.
If roots, \(r_1\) and \(r_2\text{,}\) of the characteristic equation
\begin{equation*}
ar^2+br+c=0
\end{equation*}
are real and distict, then the general solution of
\begin{equation*}
ay''+by'+cy=0
\end{equation*}
is given by
\begin{equation*}
y(x)=c_1e^{r_1x}+c_2e^{r_2x}
\end{equation*}
Theorem 94.
If roots, \(r_1\) and \(r_2\text{,}\) of the characteristic equation
\begin{equation*}
ar^2+br+c=0
\end{equation*}
are real and equal (\(r_1=r_2\)), then the general solution of
\begin{equation*}
ay''+by'+cy=0
\end{equation*}
is given by
\begin{equation*}
y(x)=c_1e^{r_1x}+c_2xe^{r_1x}
\end{equation*}
Example 95. Finding general solutions of linear DE’s with constant coefficients.
Find general solutions of the differential equations.
-
\(\displaystyle y''-2y'-3y=0\)
-
\(\displaystyle 4y''+4y'+1y=0 \)
