Questions tagged [wronskian]

Definition: The Wronskian(or Wrońskian) of $~n~$functions $y_1(x),~y_2(x),\cdots,y_n(x)~$is denoted by $~W(x)~$or,$~W(y_1,~y_2,\cdots,y_n)(x)~$and is defined to be a determinant $$W(x)~=~W(y_1,~y_2,\cdots,y_n)(x)~=~\begin{vmatrix} y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ .\\ .\\ .\\ y^{(n-1)}_1 & y^{(n-1)}_2 & \cdots & y^{(n-1)}_n \end{vmatrix}$$

Applications:

Let $y_1(x)$ and $y_2(x)$ be two real valued differentiable functions on a set $S = [a,b]$ (say). If Wronskian $W(y_1,y_2)(t_{0})$ is nonzero for some $t_{0}$ in $[a,b]$, then $y_1$ and $y_2$ are linearly independent on $[a,b]$.

If $y_1$ and $y_2$ are linearly dependent then the Wronskian $W(y_1,y_2)(t_{0})$ is zero for all $t_{0}$ in [a,b] .

Generalization:

Let $y_{1}$ and $y_{2}$ be functions of two independence variables $x_{1}$ and $x_{2}$ i.e., $y_{1} = y_{1}(x_{1} ,x_{2}) $ and $y_{2} = y_{1}(x_{1} ,x_{2}) $ for which all partial derivatives of $1^{st}$ order, $\frac{\partial y_{1}}{\partial x_{k}}$, $\frac{\partial y_{2}}{\partial x_{k}}$, $(k = 1,2)$ exists throughout the region $A$. Suppose, farther, that one of the functions, say $y_{1}$, vanishes at no point of $A$. Then if all the two rowed determinants in the matrix \begin{pmatrix} y_{1} & y_{2} \\ \frac{\partial y_{1}}{\partial x_{1}} & \frac{\partial y_{2}}{\partial x_{1}} \\ \frac{\partial y_{1}}{\partial x_{2}} & \frac{\partial y_{2}}{\partial x_{2}} \end{pmatrix} vanish identically in $A$, $y_{1}$ and $y_{2}$ are linearly dependent in $A$, and in fact $y_{2}=c y_{1}$.

References:

https://en.wikipedia.org/wiki/Wronskian

http://mathworld.wolfram.com/Wronskian.html

"Green, G. M., Trans. Amer. Math. Soc., New York, 17, 1916,(483-516)".