# Tagged Questions

24 views

### Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t),$$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
46 views

### Linearizing a nonlinear system of ODE about an equilibrium

Since the method below is probably correct, and correctness is potentially irrelevant to my ability to do what I want to learn. Assume below is correct. ...
42 views

### Differential Equations and Eigenvalues

I have the following system of differential equations: \left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. Which corresponds to the following ...
49 views

### Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$  It's a ...
Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ... 3answers 34 views ### Determinant of solution matrix Let \phi(t) be a solution matrix. Show that$$\det\phi(t)=\det\phi(t)\exp\int_{t_0}^t\sum_{j=1}^na_{jj}(s)\,ds.$$I know that [\det\phi(t)]'=\sum_{j=1}^na_{jj}(t)\det\phi(t), but I am not how to ... 1answer 82 views ### Find the determinant of a solving matrix I have such ODE:$$\frac{dy}{dt}=\begin{pmatrix} \sin^2t & e^{-t} \\ e^t & \cos^2t \end{pmatrix} y=A(t)y(t)$$and let M(t,1) be the solving matrix (a matrix whose columns ... 2answers 70 views ### Wronskian determinant and Linear dependence I was trying to show that if functions f and g defined on interval I are linearly dependent then the Wronskian determinant is zero. Suppose f, g \in I and f g are linearly dependent, then \forall ... 2answers 74 views ### how to compute the determinant of a linear map Let V be the vector space of m\times n matrices over a field F. Fix an m\times m matrix A and an n\times n matrix C, and consider the map \phi: V\longrightarrow V defined by ... 2answers 346 views ### The trace-determinant plane, classification of equilibria of differential equations What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ... 1answer 201 views ### Find the Wronskian of the Functions [closed] Find the Wronskian of the functions f(t)=6e^t\sin{t} and g(t)=e^t\cos(t). Simplify your answer. please list out all steps as simple as possible thank you 2answers 614 views ### How do I show the Wronskian of (J_{a}(x),Y_{a}(x)) = \dfrac {2} {\pi x} Based of using my undergrad class notes. I know that the wronskian of (J_{a}(x),Y_{a}(x)) is  W(J_{a}(x),Y_{a}(x)) = \left| \begin{matrix} J_{a}(x) & Y_{a}(x) \\ J_{a}'(x) & ... 1answer 92 views ### Characteristic equation for 2-nd order ODE Given a differential equation \dot x = Ax, x \in \mathbb{R}^n we define its characteristic equation as \chi(\lambda) = \det (\lambda I - A). Consider now the second order ODE$$ \ddot x + A x ...
Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if  \det\left( \begin{array}{ccccc} f_1 & f_2 & ...