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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13
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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 ...
13
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1answer
622 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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3answers
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 ...
7
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1answer
215 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 ...
5
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1answer
251 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 ...
5
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2answers
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| \...
5
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2answers
198 views

Linear independence and the Wronskian

Suppose I have two linearly independent solution vectors \begin{bmatrix}x_1,_1(t)\\x_1,_2(t)\end{bmatrix} and \begin{bmatrix}x_2,_1(t)\\x_2,_2(t)\end{bmatrix} If I take the Wronskian of these 2 ...
4
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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 ...
4
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1answer
112 views

a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $$x^2y'' + y' + (\sin x)y = 0$$ satisfying $$y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $$. I worked like following: ...
4
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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 ...
3
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1answer
181 views

Wronskian which is zero at one point

For the ODE $$xy''-(x+2)y'+2y=0~,$$ which has solutions $y_1 = e^x$ and $y_2 = x^2+2x+2$, the Wronskian is $W=-e^x x^2$. As per the known theorem, Wronskian is either identically zero (i.e. zero for ...
3
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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 ...
3
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2answers
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 ...
3
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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 ...
3
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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 ...
3
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1answer
2k views

Using the Wronskian and finding a general solution to a system of ODEs

For the system $$x' = \left[ \begin{array}{cccc} 2&6\\3&-1 \end{array} \right]x$$ with solutions $$x_1 = \left[ \begin{array}{cccc} 2e^{5t}\\e^{5t} \end{array} \right]\qquad \text{and}\quad ...
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0answers
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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2answers
1k views

Differential Equations- Wronskian Fails?

I was doing a problem where the goal was to find whether two functions: $$f(x) = \sin(2x) ,~~~~~ \text{and}~~~~~~ g(x) = \cos(2x)$$ are linearly independent or not using the wronskian. The problem ...
2
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2answers
514 views

Wronskian zero with linearly independent solutions

Any ideas how to go about proving this? Functions $\phi(x)$ and $\psi(x)$ are linearly independent on the interval $[\alpha,\beta]$, but their Wronskian $W(\phi,\psi)=0$ for some $x\in [\alpha, \beta]...
2
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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(-...
2
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2answers
415 views

If Wronskian is zero at some point in an interval, then the Wronskian is zero at all points in the interval

Suppose that $y_1$ and $y_2$ are solutions to the homogeneous DE $$y^{\prime \prime}+p(x)y^{\prime}+q(x)y=0$$ on $I$ and assume that for some $x_0 \in I$, we have $$W(y_1,y_2)(x_0)>0~.$$ Show that $...
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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, ...
2
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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???? ...
2
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3answers
209 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 ...
2
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1answer
947 views

The Wronskian of vector valued functions vs. the Wronskian of real valued functions.

Case I: When discussing, for example, two solutions $\phi_1(t)$ and $\phi_2(t)$ of a second order homogeneous ode the Wronskian $$ W[\phi_1,\phi_2](t)=\begin{vmatrix}\phi_1(t)&\phi_2(t)\\\phi_1'(t)...
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2answers
1k views

Homogeneous second-order differential equation with constant Wronskian

Problem Prove that if the Wronskian of any two solutions of differential equation $y''+p(x)y'+q(x)y=0$ is constant, then $p(x)$ is zero. My attempt. : Let $y_1$ and $y_2$ be two solutions of given ...
2
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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''-...
2
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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 ...
2
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1answer
283 views

Clarification on the Wronskian

I am getting a few "contradicting" conclusions from the Wronskian and I just wanted to clarify, but assume $y_1, y_2$ are two solutions to a second order differential equation that is homogeneous. ...
2
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2answers
864 views

Wronskian of two differential equation solutions

Let $f$ and $g$ be the solutions of the homogeneous linear equation: $$y'' + p(x)y' + q(x)y = 0$$ and $p(x)$ and $q(x)$ are continuous in segment $I$. Is it true, that if the wronskian of $f$ and $...
2
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1answer
48 views

ODE, extension of the Wronskian

I asked this question exactly about a month ago. Thought of asking again. I've answered part (a) using the definition of the wronskian $$W(t)=W(t_0)\exp\left(\int^t_{t_0} \mathrm{trace}(A(s))ds\right)...
2
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1answer
87 views

How does the Wronskian Predict DE Solutions

This is the question that was asked: Does the Wronskian of solutions of a linear homogeneous DE evolve in $t$ in any reasonable way? ($t$ being a variable.) What can you say about the Wronskian ...
2
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1answer
159 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) & f'...
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3answers
386 views

Is there any alternative to Wronskian?

Calculating a Wronskian is a very painful process, especially for higher order differential equations. Actually, I'm trying to solve a 4th order non-homogeneous linear differential equation. Consider ...
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3answers
912 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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2answers
234 views

Linear independence, yet Wronskian is zero?

Why is the following set linearly independent for all x on ($-\infty$, $\infty$)? $$\{1+x, 1-x, 1-3x\}$$ The Wronskian is $0$, but Wolfram Alpha says it is still linear independent? Why is this? ...
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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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2answers
1k views

How to find Wronskian of this ordinary differential equation problem

I know how to find the Wronskian if solutions are available. And if I can solve the problem. I don't know how to solve this problem. Is there a way to find the Wronskian of this problem without ...
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3answers
155 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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1answer
493 views

Really Confusing Question on Wronskian and ODEs

The following question was given to as in a recent assignment, namely Let $$y''+p(t)y'+q(t)y=0~.$$ It is given that $(1+t)^2$ is a solution to this differential equation, and that the wronskian of ...
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1answer
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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2answers
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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2answers
535 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 ...
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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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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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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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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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1answer
571 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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1answer
401 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
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(...