# 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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### 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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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:$$W=\begin{bmatrix}2+x&2+\... 0answers 14 views ### There cannot be two linearly independent eigenfunctions in a Sturm–Liouville problem Consider the Sturm–Liouville problem y'' + [λp(x) − q(x)]y = 0, α_1y(a) + α_2y'(a) = 0, β_1y(b) + β_2y'(b) = 0, with p(x) and q(x) continuous, p(x) > 0 on [a, b], and α_1β_1 > ... 2answers 36 views ### Wronskian and Homogenous Equation I am given three functions and an interval which I have to prove linear independence.$$\begin{align}f_1(x)&=x \\ f_2(x)&=x^2 \\ f_3(x)&=4x-3x^2\end{align}$$This is when I ask you to ... 1answer 19 views ### Showing linear independence without Wronskian So if (I,x_1) and (I,x_2) with$$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$$where p,q are continuous and we ... 0answers 27 views ### Determine if the hyperbolic and exponential functions are linearly dependent or independent The given set of functions is$$[ \sinh(x),\space\cosh(x), e^x]$$\begin{bmatrix} \sinh(x) \space \cosh(x) \space e^{-x}\\ \cosh(x) \space \space \sinh(x) \space -e^{-x}\\ \sinh(x) \space \cosh(x) \... 1answer 29 views ### Abel's Theorem Contradiction? A homework problem I assigned said: The Wronskian of a second order linear ODE is given by:$$ (x-1)e^x. $$It then asks whether the functions are linearly independent on the whole real line: yes, it ... 2answers 31 views ### Linearly independent solutions of a differential? Considering the differential equation 2x^2y''+3xy' - y = 0. Determine r\in\textbf{R}_{>0}, such that  y_2(x)=x^r be linearly independent of y_1(x)= 1/x . I know that I must calculate W(... 1answer 21 views ### Is Wronskian infinitely times differentiable? If y_1, y_2 are two solutions of the differential equation a_{0}(x)y'' + a_{1}(x)y' +a_{2}(x)y = 0 where a_{0},a_{1} a_{2} are continuous and a_{0}(x) \ne 0 Then the wronskian W of y_1,... 1answer 36 views ### Wronskian of x|x| and x^2. Wikipedia says wronskian of x|x| and x^2 is identically zero. But it is not LD. I know why these two are LI and not LD. since x|x| is not differentiable function,how to find their wronskian???? ... 1answer 40 views ### If the differential equation t^2 y'' - 2y' + (3 + t)y = 0 has y_1 and y_2 as a fundamental set of solutions… If the differential equation t^2 y'' - 2y' + (3 + t)y = 0 has y_1 and y_2 as a fundamental set of solutions and if W(y_1, y_2)(2) = 3, find W(y_1, y_2)(4). Is it possible for me to solve ... 1answer 27 views ### Substituting t_0 in Wronskian to solve 2nd order non homogeneous ode? Hey everyone I need some help understanding how to use Lagrange method and specifically the Wronskian used in it to solve 2nd order non homogeneous ode. Let :$$y''(t)+a_1y'(t)+a_2y(t)=f(t)$$I ... 1answer 53 views ### Linear independence when writing a function as a sum of functions. Consider splitting a two-variable function into a sum of products of one-variable functions like this:$$f(x,y) = \sum_{i=1}^n g_i(x) \cdot h_i(y)$$Such a decomposition is called irreducible if the ... 1answer 57 views ### How to prove linear combination solves homogeneous equation I have a third-order homogeneous linear differential equation:$$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 ... 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 ... 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''-... 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 ... 0answers 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)}, ... 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. 0answers 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 ... 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 \... 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}. 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 ... 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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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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### 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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### 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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### 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 .
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 ...
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 ...