| Version 9 |
Version 8 |
| Given functions $f_1, f_2, \dotsc, f_n$, then the \emph{Wronskian determinant} (or simply the Wronskian) $W(f_1, f_2, f_3, \dotsc, f_n)$ is the determinant of the square matrix |
Given functions $f_1, f_2, \dotsc, f_n$, then the \emph{Wronskian determinant} (or simply the Wronskian) $W(f_1, f_2, f_3, \dotsc, f_n)$ is the determinant of the square matrix |
| \[ |
\[ |
| W(f_1, f_2, f_3, \dotsc, f_n) = \left\lvert\begin{array}{@{}ccccc@{}} |
W(f_1, f_2, f_3, \dotsc, f_n) = \left\lvert\begin{array}{@{}ccccc@{}} |
| f_1 & f_2 & f_3 & \cdots & f_n\\ |
f_1 & f_2 & f_3 & \cdots & f_n\\ |
| f_1' & f_2' & f_3' & \cdots & f_n'\\ |
f_1' & f_2' & f_3' & \cdots & f_n'\\ |
| f_1'' & f_2'' & f_3'' & \cdots & f_n''\\ |
f_1'' & f_2'' & f_3'' & \cdots & f_n''\\ |
| \vdots & \vdots & \vdots & \ddots & \vdots\\ |
\vdots & \vdots & \vdots & \ddots & \vdots\\ |
| f_1^{(n-1)} & f_2^{(n-1)} & f_3^{(n-1)} & \cdots & f_n^{(n-1)}\\ |
f_1^{(n-1)} & f_2^{(n-1)} & f_3^{(n-1)} & \cdots & f_n^{(n-1)}\\ |
| \end{array}\right\rvert |
\end{array}\right\rvert |
| \] |
\] |
| where $f^{(k)}$ indicates the $k$th derivative of $f$ (not exponentiation). |
where $f^{(k)}$ indicates the $k$th derivative of $f$ (not exponentiation). |
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| The Wronskian of a set of functions $F$ is another function, which is zero over any interval where $F$ is linearly dependent. Just as a set of vectors is said to be linearly dependent when there exists a non-trivial linear relation between them, a set of functions $\{f_1, f_2, f_3, \dotsc, f_n\}$ is also said to be dependent over an interval $I$ when there exists a non-trivial linear relation between them, i.e., |
The Wronskian of a set of functions $F$ is another function, which is zero over any interval where $F$ is linearly dependent. Just as a set of vectors is said to be linearly dependent when there exists a non-trivial linear relation between them, a set of functions $\{f_1, f_2, f_3, \dotsc, f_n\}$ is also said to be dependent over an interval $I$ when there exists a non-trivial linear relation between them, i.e., |
| \[ |
\[ |
| a_1 f_1(t) + a_2 f_2(t) + \dotsb + a_n f_n(t) = 0 |
a_1 f_1(t) + a_2 f_2(t) + \dotsb + a_n f_n(t) = 0 |
| \] |
\] |
| for some $a_1, a_2, \dotsc, a_n$, not all zero, at any $t \in I$. |
for some $a_1, a_2, \dotsc, a_n$, not all zero, at any $t \in I$. |
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| Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solutions of a second-order differential equation are independent, we may use the Wronskian. |
Therefore the Wronskian can be used to determine if functions are independent. This is useful in many situations. For example, if we wish to determine if two solutions of a second-order differential equation are independent, we may use the Wronskian. |
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|
| \paragraph{Examples} |
\paragraph{Examples} |
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| Consider the functions $x^2$, $x$, and $1$. Take the Wronskian: |
Consider the functions $x^2$, $x$, and $1$. Take the Wronskian: |
| \[ |
\[ |
| W = \left\lvert\begin{array}{@{}ccc@{}} |
W = \left\lvert\begin{array}{@{}ccc@{}} |
| x^2 & x & 1\\ |
x^2 & x & 1\\ |
| 2x & 1 & 0\\ |
2x & 1 & 0\\ |
| 2 & 0 & 0\\ |
2 & 0 & 0\\ |
| \end{array}\right\rvert = -2 |
\end{array}\right\rvert = -2 |
| \] |
\] |
| Note that $W$ is always non-zero, so these functions are independent everywhere. Consider, however, $x^2$ and $x$: |
Note that $W$ is always non-zero, so these functions are independent everywhere. Consider, however, $x^2$ and $x$: |
| \[ |
\[ |
| W = \left\lvert\begin{array}{@{}cc@{}} |
W = \left\lvert\begin{array}{@{}cc@{}} |
| x^2 & x\\ |
x^2 & x\\ |
| 2x & 1\\ |
2x & 1\\ |
| \end{array}\right\rvert = x^2 - 2x^2 = -x^2 |
\end{array}\right\rvert = x^2 - 2x^2 = -x^2 |
| \] |
\] |
| Here $W = 0$ only when $x = 0$. Therefore $x^2$ and $x$ are independent except at $x = 0$. |
Here $W = 0$ only when $x = 0$. Therefore $x^2$ and $x$ are independent except at $x = 0$. |
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|
| Consider $2x^2+3$, $x^2$, and $1$: |
Consider $2x^2+3$, $x^2$, and $1$: |
| \[ |
\[ |
| W = \left\lvert\begin{array}{@{}ccc@{}} |
W = \left\lvert\begin{array}{@{}ccc@{}} |
| 2x^2 + 3 & x^2 & 1\\ |
2x^2 + 3 & x^2 & 1\\ |
| 4x & 2x & 0\\ |
4x & 2x & 0\\ |
| 4 & 2 & 0\\ |
4 & 2 & 0\\ |
| \end{array}\right\rvert = 8x - 8x = 0 |
\end{array}\right\rvert = 8x - 8x = 0 |
| \] |
\] |
| Here $W$ is always zero, so these functions are always dependent. This is intuitively obvious, of course, since |
Here $W$ is always zero, so these functions are always dependent. This is intuitively obvious, of course, since |
| \[ |
\[ |
| 2x^2 + 3 = 2(x^2) + 3(1) |
2x^2 + 3 = 2(x^2) + 3(1) |
| \] |
\] |
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| Given $n$ linearly independant functions $f_1, f_2, \dotsc, f_n$, we can use |
Given $n$ linearly independant functions $f_1, f_2, \dotsc, f_n$, we can use |
| the Wronskian to construct a linear differential equation whose solution space |
the Wronskian to construct a linear differential equation whose solution space |
|
is exactly the span of these functions. Namely, if $g$ satisfies the equation
|
is exactly the span of these functiosn. Namely, if $g$ satisfies the equation
|
| \[ |
\[ |
| W(f_1, f_2, f_3, \dotsc, f_n, g) = 0, |
W(f_1, f_2, f_3, \dotsc, f_n, g) = 0, |
| \] |
\] |
| then $g = a_1 f_1(t) + a_2 f_2(t) + \dotsb + a_n f_n(t)$ for some choice of |
then $g = a_1 f_1(t) + a_2 f_2(t) + \dotsb + a_n f_n(t)$ for some choice of |
| $a_1, a_2, \dotsc, a_n$. |
$a_1, a_2, \dotsc, a_n$. |
