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When discussing, for example, two solutions $\phi_1(t)$ and $\phi_2(t)$ of a second order homogeneous ode the Wronskian $$ W[\phi_1,\phi_2](t)=\begin{vmatrix}\phi_1(t)&\phi_2(t)\\\phi_1'(t)&\phi_2'(t)\end{vmatrix} $$ of the solutions might be mentioned. Likewise, when discussing two solutions $$ \phi_1(t)=\begin{bmatrix}x_1(t)\\y_1(t)\end{bmatrix}\qquad\phi_2(t)=\begin{bmatrix}x_2(t)\\y_2(t)\end{bmatrix} $$ of a system of two first order linear ode's the Wronskian $$ W[\phi_1,\phi_2](t)=\begin{vmatrix}x_1(t)&x_2(t)\\y_1(t)&y_2(t)\end{vmatrix} $$ of the solutions might be mentioned. Now, in the first case the Wronskian is an array of derivatives, while in the second, it is an array of vector components.

As in both cases the Wronskian provides information about linear independence of the solutions, it would appear that the two "versions" (array of derivatives vs. array of components) are not really different but I am missing the connection between the two. Can anyone provide some insight into that connection?

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  1. Consider vector functions $$ y_1(t),\ldots,y_k(t) $$ with components $$ y_j(t)=(x_j(t),x_j'(t),\ldots,x_j^{(k-1)}(t)). $$ Then the system of functions $\{y_1,\ldots,y_k\}$ is linearly dependent on interval $I$ if and only if the system of functions $\{x_1,\ldots,x_k\}$ is linearly dependent on $I$.

  2. Consider an $k$-th order linear ODE with a fundamental system of solutions $\{x_1,\ldots,x_k\}$ and reduce it to $k$ first order equations with the fundamental system of solutions $\{y_1,\ldots,y_k\}$.

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