Questions tagged [wronskian]

This tag is for various questions relating to "Wronskian". In mathematics, it is a determinant used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

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23 views

There cannot be two linearly independent eigenfunctions in a Sturm–Liouville problem

Consider the Sturm–Liouville problem $y'' + [λp(x) − q(x)]y = 0$, $α_1y(a) + α_2y'(a) = 0$, $β_1y(b) + β_2y'(b) = 0$, with $p(x)$ and $q(x)$ continuous, $p(x) > 0$ on $[a, b]$, and $α_1β_1 > ...
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1answer
25 views

If a square matrix is singular then does it necessarily mean it would have a non-trivial kernel?

This question is motivated by the idea of Wronskian and independence in Differential Equation course. Let $y_1$ and $y_2$ be two functions and I thought this matrix equation perfectly sums the idea ...
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3answers
37 views

True/False: If the Wronskian of n functions vanishes at all points on the real line then these functions must be linearly dependent in R.

I know that if a set of functions are linearly dependent, then its Wronskian = 0 at all values of t in the interval. So can you conclude that if Wronskian = 0 for all values of t in the interval, ...
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1answer
13 views

Wronskian and Homogenous Equation and Absolute Value

I was given these two functions: $f_1(x)=2+x \ \text{and} \ f_2(x)=2+\lvert{x}\rvert$ An interval as well $I_0=(-\infty,\infty)$ I began to setup my Wronskian as follows:$$W=\begin{bmatrix}2+x&2+\...
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2answers
36 views

Wronskian and Homogenous Equation

I am given three functions and an interval which I have to prove linear independence. $$\begin{align}f_1(x)&=x \\ f_2(x)&=x^2 \\ f_3(x)&=4x-3x^2\end{align}$$ This is when I ask you to ...
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0answers
27 views

Determine if the hyperbolic and exponential functions are linearly dependent or independent

The given set of functions is $$[ \sinh(x),\space\cosh(x), e^x]$$ $\begin{bmatrix} \sinh(x) \space \cosh(x) \space e^{-x}\\ \cosh(x) \space \space \sinh(x) \space -e^{-x}\\ \sinh(x) \space \cosh(x) \...
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1answer
20 views

Showing linear independence without Wronskian

So if $(I,x_1)$ and $(I,x_2)$ with $$x_2(t)=x_1(t)\int_{t_0}^t\frac{1}{x_1(s)^2}e^{-\int_{t_0}^sp(r)dr}ds,\quad (t_0,t\in I)$$are solutions to$$x''+p(t)x'+q(t)x=0$$where $p,q$ are continuous and we ...
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1answer
29 views

Abel's Theorem Contradiction?

A homework problem I assigned said: The Wronskian of a second order linear ODE is given by: $$ (x-1)e^x. $$ It then asks whether the functions are linearly independent on the whole real line: yes, it ...
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0answers
292 views

Compute the Wronskian. Abel's formula and determinant difference.

This question is not about solving the ODE, but about understanding something about the Wronskian and Abel's formula. Given the equation: $y''-\frac{3}{x}y'+\frac{5}{x^2}y=3$ I get: $y_1=x^2\cos \...
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31 views

Linearly independent solutions of a differential?

Considering the differential equation $2x^2y''+3xy' - y = 0.$ Determine $r\in\textbf{R}_{>0}$, such that $ y_2(x)=x^r$ be linearly independent of $y_1(x)= 1/x $. I know that I must calculate $W(...
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1k views

How can you calculate the derivative of this Wronskian?

If $W(y_1,y_2,y_3)=\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1' & y_2' & y_3' \\ y_1'' & y_2'' & y_3'' \end{array}\right|$, how can I show that $W'(y_1,y_2,y_3)=\left| \...
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1answer
22 views

Is Wronskian infinitely times differentiable?

If $y_1, y_2$ are two solutions of the differential equation $a_{0}(x)y'' + a_{1}(x)y' +a_{2}(x)y = 0$ where $a_{0},a_{1} a_{2}$ are continuous and $a_{0}(x) \ne 0$ Then the wronskian $W$ of $y_1,...
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1answer
36 views

Wronskian of $x|x|$ and $x^2$.

Wikipedia says wronskian of $x|x|$ and $x^2$ is identically zero. But it is not LD. I know why these two are LI and not LD. since x|x| is not differentiable function,how to find their wronskian???? ...
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1answer
40 views

If the differential equation $t^2 y'' - 2y' + (3 + t)y = 0$ has $y_1$ and $y_2$ as a fundamental set of solutions…

If the differential equation $t^2 y'' - 2y' + (3 + t)y = 0$ has $y_1$ and $y_2$ as a fundamental set of solutions and if $W(y_1, y_2)(2) = 3$, find $W(y_1, y_2)(4)$. Is it possible for me to solve ...
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1answer
150 views

Linear Independence of 2 Functions Via Wronskians

"Determine whether $f(x) = x^3, g(x) = x^2|x|$ are linearly independent on the real line." My Work: Independent if Wronskian $(f,g) ≠ 0$ for all points (Note: the square brackets are a determinant) ...
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2answers
2k 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) & Y_{a}'(...
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1answer
225 views

Finite family of analytic functions linearly dependent if and only if Wronskian is $~0~$

I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalize ...
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15k views

Linear independence of function vectors and Wronskians

I am taking a course in ODE, and I got a homework question in which I am required to: Calculate the Wronskians of two function vectors (specifically $(t, 1)$ and $(t^{2}, 2t)$). Determine in what ...
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1answer
484 views

Wronskian of n functions and linear in/dependence

The Wronskian of $n$ functions $y_1(x),\ldots,y_n(x)$ that are $\mathcal{C}^{n-1}$ in some interval $I$ is defined to be the determinant of the matrix \begin{pmatrix} y_1(x) & ⋯ & y_n(x) \\ ...
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1answer
1k views

Show that the Wronskian of solutions of $y''+p(x)y'+q(x)y=0$ satisfies $\frac{dW}{dx}+pW=0$

So I am given: $\{y_1(x),y_2(x)\}$ is a fundamental solution set of the ODE: $$y''+p(x)y'+q(x)y=0$$ I need to show that the Wronskian $W(y_1,y_2)$ satisfies the ODE $\frac{dW}{dx}+pW=0$ and hence, $W(...
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1answer
191 views

Show that the Wronskian of $x^{(1)},x^{(2)},x^{(3)}$ is identically zero or else never vanishes.

My task is to prove that: If $x^{(1)},x^{(2)},x^{(3)}$ are solutions of $X'=A(t)X$ on some interval $I$, show that the Wronskian of $x^{(1)},x^{(2)},x^{(3)}$ is identically zero or else never ...
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1answer
320 views

Wronskian of Airy functions.

I am trying to show that that the Airy functions defined below satisfy: $W[Ai(x),Bi(x)]=1/\pi$. $$Ai(x)=\frac{1}{\pi} \int_0^\infty \cos(t^3/3+xt)dt$$ $$Bi(x)=\frac{1}{\pi}\int_0^\infty \bigg[ \exp(-...
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1answer
413 views

Does vanishing of wronskian of solutions at point $\implies$ solutions are linearly dependent?

Let $u$ and $v$ be two solutions of $y''+P(x)y'+Q(x)y=0$,Let $W(u,v)$ denote the wronskian of $u$ and $v$ then $W(u,v)$ vanishes at a point $x_0\in[a,b]\implies u$ and $v$ are linearly dependent $W(...
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343 views

Wronskian of second order differential equation $ty'' − (t + 1)y' − y = 0$.

Find a Wronskian of two solutions of $$ty'' − (t + 1)y' − y = 0, ~~~t > 0$$ provided $W[y_1, y_2](1) = 1$. Answer: $W [y_1, y_2] (t) = te^{t−1}$ I am unsure of how they got the answer, am I ...
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0answers
26 views

When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution?

When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution? Basically, do I have to repeat the substitution with the ...
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1answer
27 views

Substituting $t_0$ in Wronskian to solve 2nd order non homogeneous ode?

Hey everyone I need some help understanding how to use Lagrange method and specifically the Wronskian used in it to solve 2nd order non homogeneous ode. Let : $$y''(t)+a_1y'(t)+a_2y(t)=f(t)$$ I ...
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1answer
57 views

How to prove linear combination solves homogeneous equation

I have a third-order homogeneous linear differential equation: $$A_3(u) f^{\prime\prime\prime} + A_2(u) f^{\prime\prime} + A_1(u) f^\prime + A_0(u) f = 0,$$ with three linearly-independent solutions $...
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1answer
53 views

Linear independence when writing a function as a sum of functions.

Consider splitting a two-variable function into a sum of products of one-variable functions like this: $$f(x,y) = \sum_{i=1}^n g_i(x) \cdot h_i(y)$$ Such a decomposition is called irreducible if the ...
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1answer
41 views

Clarification regarding the Wronskian in differential equations

Given DE is: $y''+ p(x)y'+ q(x)y = 0$ It's given in my book that if $p(x),q(x)$ are continuous on open interval I, they are linearly dependent if and only if their Wronskian W=0 at some point in the ...
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1answer
34 views

Solve $y''-3y'-4y=3\mathit{e}^{2t}$ using variation of parameters formula.

Solve $y''-3y'-4y=3\mathit{e}^{2t}$ using variation of parameters formula. I just want to know whether I am on the right track with my solution. I started by solving the differential equation $y''-...
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1answer
25 views

any two solution of the equation $y''+p(x)y'+q(x)y=0$, $p(x)$ and $q(x)$ are continuous on $(a,b)$ and $x\in (a,b)$ are linearly dependent

Any two solution of the equation $y''+p(x)y'+q(x)y=0$, $p(x)$ and $q(x)$ are continuous on $(a,b)$ and $x\in (a,b)$ are linearly dependent if (a) they have common zero in $(a,b)$ (b) they have a ...
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919 views

Prove $y=\sin(t^2)$ cannot be a solution on an interval containing $t=0$ of an equation $y'' + p(t)y'+q(t)y=0$

I want to prove that $y=\sin(t^2)$ cannot be a solution on an interval containing $t=0$ of an equation $$y'' + p(t)y'+q(t)y=0$$ using the Wronskian and Abel's formula. Let $y_{1}$ and $y_{2}=\sin(t^...
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34 views

Is Wronskian a Line Bundle for Riemann surfaces?

Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...
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2answers
42 views

Are the functions $\sin(x)$ ; $\sin^2(x)$ ; … ; $(\sin(x))^{2017}$ linearly independent?

We have the functions $\sin(x)$ ; $\sin^2(x)$ ; ... ; $(\sin(x))^{2017}$ defined on $\Bbb R$. I tried to calculate the Wronskian, but it doesn't seem to do help in any way.
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1answer
54 views

Why is it that solutions $x_1 \text{ and } x_2 \text{ to a differential equation are linearly dependent } \iff W(x_1,x_2)(t_0)=0$

Can someone explain from where the aforementioned fact is derived? By the way $W = \text { Wronskian determinant}.$
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16 views

What are the ways to find a function $g(x)$ if the Wronskian of two functions, one f(x) and g(x) is known and value of g(x) at a point is known.

So, what is the way to find a function $g(x)$ if $f(x)$ is known, and the Wronskian of these two functions is known as well. Also a value of $g$ at certain $x$ is known. How do I get to determine what ...
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0answers
19 views

Determine whether the functions are linearly independent at the interval $I$. (In the context of differential equations)

So, here are two functions $x_1(t)=\cos(2t)-1 \text{ and } x_2(t)=\sin^2(t), I = R$. $C_1x_1(t)+C_2x_2(t)=0$ Here I am stuck, because I try to find the values of $t$ such that I can express $C_1 \...
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1answer
36 views

Wronskian and Linear Dependence

Let $y_1$, $y_2$ be two solutions of a homogeneous linear second order differential equation $y^{''}$ + $p(t)y^{'}$ + $q(t)y$ = $0$ over the interval $\alpha$ < $t$ < $\beta$. Prove that ...
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0answers
10 views

The total wronskian determinant

Given a sequence of function, $$F=\{f_1(x,t),f_2(x,t),f_3(x,t),\cdots,f_m(x,t)\},$$ we define the total Wronski determinant of this set of functions as $$W(F)=\det\begin{vmatrix}F\\D_xF\\D_tF\\\vdots\\...
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1answer
74 views

Question about Wronskian.

I'm studying Linear Homogeneous Equations and Wronskian, and I saw Theorems that Let $S$ be the set of all solutions of $y^{''}$ + $py^{'}$ + $qy$ = $0$, where $p(t)$ and $q(t)$ are continuous in ...
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2answers
180 views

A more elegant way of computing this Wronskian?

As I was working on my differential equation homework this week I came across this problem: Let $y^{(4)} + 16y=0$. Compute the Wronskian of four linearly independent solutions. It's rather ...
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1answer
881 views

Suppose that the Wronskian of any 2 solutions of $\frac{d^2y}{dt^2}+p(t)\frac{dy}{dt}+q(t)y=0$ Prove that P(t)=0.

Suppose that the Wronskian of any 2 solutions is constant of $\frac{d^2y}{dt^2}+p(t)\frac{dy}{dt}+q(t)y=0$ Prove that P(t)=0. So my attempt: $$W(t)=y_1y_2'-y_1'y_2$$ So what I thought I would do is ...
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1answer
47 views

Which among the following is true for a Wronskian Of a Differential Equation

Let $u$ and $v$ be two solutions of the differential Equation $y^{"} + P(x)y^{'} + Q(x)y = 0$ on $[a,b]$, Let $W(u,v)$ denote the Wronskian Of $u$ and $v$ Then (a) $W(u,v)$ vanishes at point $x_{0}...
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1answer
6k views

Proof that ODE solutions with Wronskian identically zero are linearly dependent

According to Wikipedia, if the Wronskian of two functions is always zero, then they are not necessarily linearly dependent. But it seems that if the two functions are solutions of the same ...
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1answer
31 views

Feedback regarding a solution based on a Wronskian

Function $y_1(x)$ and $y_2(x)$ are solutions to the equation $y''+\frac{1}{x}y'+(1-\frac{1}x^2)y=0$ in the open interval $(0,\infty$). It is given that the following conditions are fulfilled: $y_1(1)=...
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0answers
15 views

Numerical evaluation of fractions including spherical bessel functions with large complex argument

I need to numerically evaluate some fractions $$\frac{j_n(a)h'_n(b)-h_n(a)j'_n(b)}{j'_n(a)h'_n(b)-h'_n(a)j'_n(b)} \frac{1}{a}$$ $$\frac{j_n(a)h'_n(a)-h_n(a)j'_n(a)}{j'_n(a)h'_n(b)-h'_n(a)j'_n(b)} \...
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2answers
1k views

Construct a Second Order ODE given the fundamental solutions

I need to construct a second order linear differential equation for which $\{ \sin (x), x \sin (x) \}$ is the set of fundamental solutions. I am completely lost on this problem and have been trying ...
13
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1answer
624 views

How to prove a Wronskian identity?

The following Wronskian identity can be proved by expanding both sides and checking that two sides are the same. But how to prove it more elegantly? Let $u_1(x), u_2(x), u_3(x), u_4(x)$ be four ...
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1answer
552 views

Linear independence and the Wronskian

I want to show linear independence in the wronskian implies linear independence between the functions $f_1(x)$, $f_2(x)$, $f_3(x)$. Let $f_1(x)$, $f_2(x)$, $f_3(x)$ be real-valued functions with first ...
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0answers
133 views

Linear dependance and Wronskian determinant

I am asked to show that the fuctions $e^x, \cos(x) \text{ and } x^2$ are linearly independent. I wanted to use the Wronskian determinant in order to prove the above property. We have: $$W= \begin{...