4.3: The Wronskian
- Page ID
- 90408
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Suppose that having determined that two solutions of (4.2.1) are \(x = X_1(t)\) and \(x = X_2(t)\), we attempt to write the general solution to (4.2.1) as (4.2.2). We must then ask whether this general solution will be able to satisfy the two initial conditions given by \[\label{eq:1}x(t_0)=x_0,\quad\overset{.}{x}(t_0)=u_0.\]
Applying these initial conditions to (4.2.2), we obtain \[\begin{align}c_1X_1(t_0)+c_2X_2(t_0)&=x_0,\nonumber \\ c_1\overset{.}{X}_1(t_0)+c_2\overset{.}{X}_2(t_0)&=u_0,\label{eq:2}\end{align}\] ) which is observed to be a system of two linear equations for the two unknowns \(c_1\) and \(c_2\). Solution of \(\eqref{eq:2}\) by standard methods results in \[c_1=\frac{x_0\overset{.}{X}_2(t_0)-u_0X_2(t_0)}{W},\quad c_2=\frac{u_0X_1(t_0)-x_0\overset{.}{X}_1(t_0)}{W},\nonumber\] where \(W\) is called the Wronskian and is given by \[\label{eq:3}W=X_1(t_0)\overset{.}{X}_2(t_0)-\overset{.}{X}_1(t_0)X_2(t_0).\]
Evidently, the Wronskian must not be equal to zero \((W\neq 0)\) for a solution to exist.
For examples, the two solutions \[X_1(t)=A\sin\omega t,\quad X_2(t)=B\sin\omega t,\nonumber\] have a zero Wronskian at \(t=t_0\), as can be shown by computing \[\begin{aligned} W&=(A\sin\omega t_0)(B\omega\cos\omega t_0)-(A\omega\cos\omega t_0)(B\sin\omega t_0) \\ &=0;\end{aligned}\] while the two solutions \[X_1(t)=\sin\omega t,\quad X_2(t)=\cos\omega t,\nonumber\] with \(\omega\neq 0\), have a nonzero Wronskian at \(t=t_0\), \[\begin{aligned}W&=(\sin\omega t_0)(-\omega\sin\omega t_0)-(\omega\cos\omega t_0)(\cos\omega t_0) \\ &=-\omega.\end{aligned}\]
When the Wronskian is not equal to zero, we say that the two solutions \(X_1(t)\) and \(X_2(t)\) are linearly independent. The concept of linear independence is borrowed from linear algebra, and indeed, the set of all functions that satisfy (4.2.1) can be shown to form a two-dimensional vector space.