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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30 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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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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14 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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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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1answer
19 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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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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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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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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1answer
21 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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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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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
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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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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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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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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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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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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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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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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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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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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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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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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73 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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1answer
46 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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67 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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153 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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88 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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83 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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48 views

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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144 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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90 views

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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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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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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126 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
157 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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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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40 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
114 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
63 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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114 views

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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90 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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87 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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406 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
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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86 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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1answer
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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1answer
60 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
569 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 ...