2
votes
0answers
41 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 ...
1
vote
1answer
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 ...
1
vote
0answers
53 views

Number of zeros of Wronskian

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) & ...
0
votes
3answers
32 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 ...
1
vote
1answer
80 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 ...
1
vote
2answers
66 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 ...
3
votes
2answers
73 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 ...
1
vote
2answers
342 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 ...
-1
votes
1answer
199 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
1
vote
2answers
582 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) & ...
3
votes
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 ...
6
votes
2answers
272 views

Determinant called Grammian

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 & ...