7.1: Introduction to Linear Higher Order Equations
- Page ID
- 103555
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)If you have not had Math 410 (linear algebra), then you will need to read Appendix 11.4 before starting Chapter 7.
An \(n\)th order differential equation is said to be linear if it can be written in the form
\[\label{eq:9.1.1} y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x).\]
We considered equations of this form with \(n=1\) in Section 2.3 and with \(n=2\) in Chapter 5. In this chapter \(n\) is an arbitrary positive integer. In this section we sketch the general theory of linear \(n\)th order equations. Since this theory has already been discussed for \(n=2\) in Sections 5.1 and 5.4, we’ll omit proofs.
For convenience, we consider linear differential equations written as
\[\label{eq:9.1.2} P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=F(x),\]
which can be rewritten as Equation \ref{eq:9.1.1} on any interval on which \(P_0\) has no zeros, with \(p_1=P_1/P_0\), …, \(p_n=P_n/P_0\), and \(f=F/P_0\).
The next theorem is analogous to Theorem 5.4.1.
Suppose Equation \ref{eq:9.1.1} has continuous coefficients on \((a,b)\), let \(x_0\) be a point in \((a,b),\) and let \(k_0\), \(k_1\), …, \(k_{n-1}\) be arbitrary real numbers\(.\) Then the initial value problem
\[y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x), \quad y(x_0)=k_0,\quad y'(x_0)=k_1,\dots,\quad y^{(n-1)}(x_0)=k_{n-1}\nonumber \]
has a unique solution on \((a,b)\).
Homogeneous Equations
Equation \ref{eq:9.1.2} is said to be homogeneous if \(f(x)\equiv0\) and nonhomogeneous otherwise. Since \(y\equiv0\) is obviously a solution, we call it the trivial solution. Any other solution is nontrivial.
If \(y_1\), \(y_2\), …, \(y_n\) are defined on \((a,b)\) and \(c_1\), \(c_2\), …, \(c_n\) are constants, then
\[\label{eq:9.1.3} y=c_1y_1+c_2y_2+\cdots+c_ny_n\]
is a linear combination of \(\{y_1,y_2\dots,y_n\}\). It’s easy to show that if \(y_1\), \(y_2\), …, \(y_n\) are solutions of the homogeneous equation on \((a,b)\), then so is any linear combination of \(\{y_1,y_2,\dots,y_n\}\). (See the proof of Theorem 5.1.2.) We say that \(\{y_1,y_2,\dots,y_n\}\) is a fundamental set of solutions of the homogenous equation on \((a,b)\) if every solution on \((a,b)\) can be written as a linear combination of \(n\) linearly independent solutions \(\{y_1,y_2,\dots,y_n\}\), as in Equation \ref{eq:9.1.3}. In this case we say that Equation \ref{eq:9.1.3} is the general solution of the homogeneous equation on \((a,b)\).
The problem we have here is determining whether the set of solutions is independent or not. With only two, it is simple because we only need to see if one is a constant multiple of the other. However, once we get beyond two it's more complicated and typically requires knowledge of linear algebra, and this is where the idea of the Wronskian becomes more prominent.
Recall from section 5.1, we defined the Wronskian of \(\{y_1,y_2\}\) as the function \(W=y_1y_2'-y_1'y_2\).
\[W=\left| \begin{array}{cc} y_1 & y_2 \\ y'_1 & y'_2 \end{array} \right|=y_1y_2'-y_1'y_2\nonumber \]
which you can now see is just a determinant. Note that the first row is made up of \(y_1\) and \(y_2\) and the second is made up of their derivatives. This idea extends to the following definition.
The Wronskian of \(\{y_1,y_2\dots,y_n\}\) is the determinant
\[W=\left|\begin{array}{cccc} y_1(x)&y_2(x)&\cdots&y_n(x)\\[4pt] y'_1(x)&y'_2(x)&\cdots&y_n'(x)\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] y_1^{(n-1)}(x)&y_2^{(n-1)}(x)&\cdots&y_n^{(n-1)}(x) \end{array}\right|.\nonumber\]
Note that the first row is made up of \(y_1\), \(y_2\),...,\(y_n\), the second row is made up of their derivatives, the third row is made up of the second derivatives, etc. You will always end up with an \(n \times n\) matrix.
The next theorem is analogous to Theorem 5.1.3.
Suppose \(y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=0\) has continuous coefficients on \((a,b)\) and let \(y_1\), \(y_2\), …, \(y_n\) be \(n\) solutions on \((a,b)\). Then
- \(\{y_1,y_2,\dots,y_n\}\) is linearly independent on \((a,b)\) if and only if W\(\not\equiv0\) on \((a,b)\).
- \(\{y_1,y_2,\dots,y_n\}\) is linearly dependent on \((a,b)\) if and only if W\(\equiv0\) on \((a,b)\).
So, \(\{y_1,y_2,\dots,y_n\}\) is a fundamental set of solutions of if and only if W\(\not\equiv0\) on \((a,b)\), and \(y=c_1y_1+c_2y_2+...+y_n\) is therefore the general solution of \(y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=0\) .
The next two examples illustrate concepts that we’ll develop later in this section. You shouldn’t be concerned with how to find the given solutions of the equations in these examples. This will be explained in later sections.
The equation
\[\label{eq:9.1.5} x^3y'''-x^2y''-2xy'+6y=0\]
has the solutions \(y_1=x^2\), \(y_2=x^3\), and \(y_3=1/x\) on \((-\infty,0)\) and \((0,\infty)\). Show that \(\{y_1,y_2,y_3\}\) is linearly independent on \((-\infty, 0)\) and \((0,\infty)\). Then find the general solution of Equation \ref{eq:9.1.5} on \((-\infty, 0)\) and \((0,\infty)\).
Solution
\[W=\left|\begin{array}{cccc} x^2&x^3&{1\over x}\\[4pt] 2x&3x^2&{-1\over x^2}\\[4pt] 2&6x&{2\over x^3} \end{array}\right|=12x\nonumber\]
Therefore \(W(x)\ne0\) on \((-\infty,0)\) and \((0,\infty)\) and \(\{x^2,x^3,{1\over x}\}\) is linearly independent on \((-\infty,0)\) and \((0,\infty)\) thereby forming a fundamental set of solutions to \ref{eq:9.1.5}. Therefore, the general solution to \ref{eq:9.1.5} is \[y=c_1x^2+c_2x^3+c_3{1\over x}.\nonumber\]
The equation
\[\label{eq:9.1.9} y^{(4)}+y'''-7y''-y'+6y=0\]
has the solutions \(y_1=e^x\), \(y_2=e^{-x}\), \(y_3=e^{2x}\), and \(y_4=e^{-3x}\) on \((-\infty,\infty)\). (Verify.) Show that \(\{y_1,y_2,y_3,y_4\}\) is linearly independent on \((-\infty,\infty)\). Then find the general solution of Equation \ref{eq:9.1.9}.
Solution
\[W=\left|\begin{array}{cccc} e^x&e^{-x}&e^{2x}&e^{-3x}\\[4pt] e^x&-e^{-x}&2e^{2x}&-3e^{-3x}\\[4pt] e^x&e^{-x}&4e^{2x}&9e^{-3x}\\[4pt]e^x&-e^{-x}&8e^{2x}&-27e^{-3x} \end{array}\right|=240e^{-x}\nonumber\]
Therefore \(W(x)\ne0\) on \((-\infty,\infty)\) and \(\{e^x,e^{-x},e^{2x},e^{3x}\}\) is linearly independent on \((-\infty,\infty)\) thereby forming a fundamental set of solutions to \ref{eq:9.1.9}. Therefore, the general solution to \ref{eq:9.1.9} is \[y=c_1e^x+c_e^{-x}+c_3e^{2x}+c_4e^{3x}.\nonumber\]
General Solution of a Nonhomogeneous Equation
The next theorem is analogous to Theorem 5.4.2. It shows how to find the general solution of \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x) \) if we know a particular solution of \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x) \) and a fundamental set of solutions of the homogeneous equation \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=0\).
Suppose \(p_1(x)\),...,\(p_n(x)\), and \(f(x)\) are continuous on \((a,b).\) Let \(y_p\) be a particular solution of
\[\label{eq:9.1.10}y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x)\]
on \((a,b)\), and let \(\{y_1,y_2,...,y_n\}\) be a fundamental set of solutions of the homogeneous equation
\[\label{eq:9.1.11}y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=0\]
on \((a,b)\). Then \(y\) is a solution of \(\eqref{eq:9.1.10}\) on \((a,b)\) if and only if
\[\label{eq:9.1.12} y=y_p+c_1y_1+c_2y_2+...c_ny_n\]
where \(c_1\),\(c_2\),....,\(c_n\) are constant.
We first show that \(y\) in Equation \ref{eq:9.1.12} is a solution of Equation \ref{eq:9.1.10} for any choice of the constants \(c_1\) and \(c_2\). Differentiating Equation \ref{eq:9.1.12} twice yields
\[y'=y_p'+c_1y_1'\cdots+c_ny_n' \quad \text{and} \quad y''=y_p''+ c_1y_1''\cdots+c_ny_n'', \nonumber\]
so
\[\begin{align*} y''+p(x)y'+q(x)y&=(y_p''+c_1y_1''\cdots+c_ny_n'') +p(x)(y_p'+c_1y_1'\cdots+c_ny_n') +q(x)(y_p+c_1y_1\cdots+c_ny_n)\\ &=(y_p''+p(x)y_p'+q(x)y_p)+c_1(y_1''+p(x)y_1'+q(x)y_1) \cdots+c_n(y_n''+p(x)y_n'+q(x)y_n)\\ &= f+c_1\cdot0\cdots+c_n\cdot0=f,\end{align*}\]
since \(y_p\) satisfies Equation \ref{eq:9.1.10} and \(y_1\) and \(y_2\) satisfy Equation \ref{eq:9.1.11}.
Now we’ll show that every solution of Equation \ref{eq:9.1.10} has the form Equation \ref{eq:9.1.12} for some choice of the constants \(c_1\) and \(c_2\). Suppose \(y\) is a solution of Equation \ref{eq:9.1.10}. We’ll show that \(y-y_p\) is a solution of Equation \ref{eq:9.1.11}, and therefore of the form \(y-y_p=c_1y_1 \cdots +c_ny_n\), which implies Equation \ref{eq:9.1.12}. To see this, we compute
\[\begin{align*} (y-y_p)''+p(x)(y-y_p)'+q(x)(y-y_p)&=(y''-y_p'')+p(x)(y'-y_p') +q(x)(y-y_p)\\ &=(y''+p(x)y'+q(x)y) -(y_p''+p(x)y_p'+q(x)y_p)\\ &=f(x)-f(x)=0,\end{align*}\]
since \(y\) and \(y_p\) both satisfy Equation \ref{eq:9.1.10}.
We say that Equation \ref{eq:9.1.12} is the general solution of \(\eqref{eq:9.1.10}\) on \((a,b)\).
The next theorem is analogous to Theorem 5.4.3.
Suppose for each \(i=1,\) \(2,\) …, \(r\), the function \(y_{p_i}\) is a particular solution of \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f_i(x) \) on \((a,b).\) Then
\[y_p=y_{p_1}+y_{p_2}+\cdots+y_{p_r} \nonumber\]
is a particular solution of
\[ y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f_1(x)+f_2(x)+\cdots+f_r(x) \nonumber\]
on \((a,b).\)
We’ll apply Theorems 7.1.6 and 7.1.7 throughout the rest of this chapter.