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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27 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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17 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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31 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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38 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
24 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
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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50 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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38 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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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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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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33 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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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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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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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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53 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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33 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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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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54 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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39 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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64 views

System Of ODEs - Wronskian

I am reading about the Wronskian but I find a lot of conflicts in the theory, as you can see in the pictures below. In some sources Wronskian includes the linear independent solutions and in others ...
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129 views

How to know if a function is linearly independent or dependent?

Original Problem: Determine if the set of functions $$\{ y_1(x),y_2(x),y_3(x) \} = \{x^2, \sin x, \cos x \}$$ is linearly independent. I understand I have to use the Wronskian method, but how would ...
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86 views

How to show that a given differential equation has no solution?

Let $h(x)=1+x$ and $z(x)=e^x$ be two solutions of $$y''(x)+P(x)y'+Q(x)y(x)=0$$ Then the set of conditions for which the DE has no solution is $y(0)=2$, $y'(0)=1$ $y(1) =0$ , $y'(1)= 1$ The answer ...
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77 views

Second Kummer Function

Background: I am trying to solve the radial Schroedinger equation in the form: \begin{align} \frac{\partial^2 P}{\partial r^2} + 2 \left(E + \frac{Z}{r} - \frac{l(l+1)}{2r^2}\right) P = 0 \end{align} ...
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Linear Independence of Expressions and Derivatives

To remain square, the Wronskian picks up an extra derivative whenever a new expression is added. Finding whether $n$ expressions are linearly independent requires taking $n - 1$ derivatives of each. ...
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104 views

Wronskian, Fundamental Matrix, and Elementary Row/Column Operations

The terms "Jacobian" and "Hessian" can both either refer to a matrix, or the determinant of that matrix. However, the term "Wronskian" seems only to refer to a determinant. My understanding from ...
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Why Wronskian works for DEs, but in general it doesn't show independence

For a set of functions $f_1,f_2,...f_n$, if their Wronskian determinant is identically zero $W(f_1,...f_n)(x) = 0$ for all $x$ in some interval $I$ we can't conlcude that these functions are linearly ...
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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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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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111 views

Problem regarding Wronskian

I have got a problem from Wronskian..I am a first reader of Differential Equation. Can anyone please help me to solve this problem? Attempt: I know that if $n$ number of $n-1$ differentiable ...
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1answer
121 views

First derivative of the Wronskian?

I'm not sure how to find the first derivative of the Wronskian. I have the equation of the Wronskian for two functions where I only use the functions and their first derivatives. I have the following:...
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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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39 views

Clarification of Wronskian

Let's say I have an interval consisting of 10 elements and I calculate the Wronskian of the given functions. Out of the interval of 10 elements, substitution of 9 of the elements in the equation ...
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1answer
89 views

Principle of Superposition and Wronskian

Assume that $p$ and $q$ are continuous and that the functions $y_1$ and $y_2$ are solutions of the differential equation $$y''+p(t) y'+q(t)y=0$$ an open interval $I$. Prove that if $y_1$ and $y_2$ are ...
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1answer
59 views

Wronskian is not defined

Suppose a general question: What does it mean that the Wronskian at a certain point is not even defined? take for example two solutions for a second order ODE: $$ y_1(x)=\frac{1}{x^{3}} ~~~~\text{and} ...
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Wronskian of set of solutions to $2$nd order ODE vs systems of two $1$st order ODEs

Let $x_1 = y$ and $x_2 = y'$ and convert the ODE $$y'' + p(t)y'+ q(t)y = 0$$ to a system of two first-order ODEs in $x_1$ and $x_2$. Then show that if $x_1$ and $x_2$ form a fundamental set of ...
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81 views

Linear dependence and linear independence of functions in linear algebra

I am trying to understand linear dependence and linear independence of real valued functions on a set. Say S. I want to know that using wronskain how can we say that a set S of functions is linearly ...
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81 views

Wronskian is infinite

Given a differential equation $$xy''+y'+xy=0~.$$ I have to find the number of linearly independent solutions about $x=0$. My attempt is to find the Wronskian at $x=0$ which is infinity. I cannot ...
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372 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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1answer
306 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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84 views

Wronskian Problem

My approach for this question, I consider $a$ and $b$ as some specific real numbers and then, after solving the differential equation, I get $y_i$, $i=1,2$ and as a function of $t$ and go on solving ...
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104 views

wronskian for showing linear independent [closed]

Show that the functions $~f_1(x)=e^x,~ f_2(x)=xe^x,~ f_3(x)=x^2 e^x~$ are linear independent by using Wronskian .
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58 views

Wronskian for functions

I understand that if $~y_1,y_2,\cdots,y_n~$ are solutions of a normalized homogenous linear differential equation in $~I~$, then the wronskian of the solutions is always $~0~$ or never $~0~$ for every ...
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1answer
526 views

Dependence implies Wronskian being zero

From Wikipedia: If the functions $f_i$ are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian ...
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326 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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1answer
373 views

Wronskian identically zero or never zero?

Consider the equation $$y''- \dfrac{y'}{x} = 0~.$$ Solution is $$y=Cx^2 + d~.$$ The Wronskian of $x^2$ and $1$ turns out to be $-2x$ which is zero at $x = 0$ and non zero elsewhere. But the ...
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1answer
141 views

Wronskian for multivariate functions

I've been reading about the wronskian and I got stuck in the following: Suppose we are given a multivariate function describing e.g. a plane: $z = m_1 x + m_2 y + b$. How is the wronskian computed? ...
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242 views

Intuition of the Wronskian

I've got a question regarding the intuition of a Wronskian, in the following sense: The intuition for the determinant of a square $n \times n $-matrix is that it represents the area/(hyper-)volume ...
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1answer
78 views

Wronskian of second order DE?

Let $y_1, y_2$ be two solutions of $$y''(t)+ay'(t)+by(t)=0 ~~~~~\text{for}~~~ t\in \Bbb{R}~~~~~ \text{and}~~~ y(0)=0$$ where $a,b$ are real constants. Let $W$ be the Wronskian of $y_1$ and $y_2$, ...
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3answers
206 views

Find The Wronskian Of The Following ODE

Let $y_1$ and $y_2$ be two solutions of the problem, $$\begin{align} y''(t)+ay'(t)+by(t)=0,t\in \Bbb R\\ y(0)=0\;\;\;\;\;\;\;\;\; \end{align}$$ where $a$ and $b$ are real constants. Let $W$ be the ...
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1answer
184 views

Relationship between the Wronskian and the Gramian

Is it possible to draw some parallels between the Wronskian and the Gram matrix? Could they be used for solving the same problem? What is the principal difference between them? The Gram matrix of a ...