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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Proving solutions of $y''+p(x)y'+q(x)y=0$ to be linearly independent

When studying Elementary Differential Equations by William, I found trouble understanding Theorem 5.1.5 It says the two solutions are linearly independent iff their Wronskian is never zero, but I ...
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Wronskian & Linear independence of functions

We usually study wronskian concerned with solutions of ODE, but using this idea to check linear dependence of any two differentiable functions seems Okay because it is the determinant of a Matrix ...
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Find general solution of Differential equation if you know three solutions. Is there exist general solution if Wronskian is zero? [closed]

Find general solution of Differential equation if you know three solutions. I tried to solve this problem, however I have a question about the Wronskian. Three particular solutions are 1, $x$, ${x^2}$....
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Derivative of a rational function on a curve with respect to a nonconstant function

Let $C$ be a smooth projective curve over an algebraically closed field $k$, and let $k(C)$ be its function field. For elements $f_1,...,f_n \in k(C)$ and a nonconstant $t \in k(C) \setminus k$, we ...
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Is this set of functions linearly dependent or independent? [duplicate]

The given functions are solutions to a differential equation \begin{equation*} y_1(x)=\cos(2x),y_2(x)=1,\;y_3(x)=\cos(x) \end{equation*} I need help determining if the set of functions are linearly ...
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Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation, has only simple zeroes.

Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation, $$a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$$ $\forall\ x \in I$ has only simple zeroes. Where point $...
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How can we use the initial condition $y(0)=0$, the solution is not even defined at $0$?

The question states: Consider the differential equation $x^2y''+xy'-y=0$. If $y_1$ and $y_2$ are two linearly independent solutions to the differential equation then choose the incorrect: (1) $W(y_1,...
math student's user avatar
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Why A and B need not be differentiable...?

The method of variation of parameters to solve differential equations $y''+p(x)y'+Q(x)y=R(x)\ x\in I, P,Q,R$ are continuous functions for every $x\in I$ seeks a particular solution of the form $y(x)=...
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Formula for Derivative of Wronskian with Trace

Let $W$ be the Wronski matrix of a fundamental system of the homogeneous linear differential equation system $$ y^{(k)}+A_{k-1}(x) y^{(k-1)}+\ldots+A_1(x) y^{\prime}+A_0(x) y=0, $$ where the $A_j$ are ...
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Wronskian - independent solutions

Question based on Example sheet 3, Problem 2 from here: https://dec41.user.srcf.net/notes/IA_M/differential_equations_eg.pdf Could somebody tell me if there's a mistake in the initial conditions for $...
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Why the solution is wrong? $(x^2+x)\frac{d^2y}{ dx^2}+(2-x^2)\frac{ dy}{ dx}-(2+x)y=x(1+x)^2$

$$ (x^2+x)\frac{\mathrm d^2y}{\mathrm dx^2}+(2-x^2)\frac{\mathrm dy}{\mathrm dx}-(2+x)y=x(1+x)^2\tag*{}$$ This is the D.E. I'm trying to solve via the method of variation of parameter. As per this ...
Nothing special's user avatar
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Differential equation of the form: $y'' - \frac{1}{(1+x^2)^2} y=-\frac{Ax}{\sqrt{1+x^2}}$

When attempting this problem on physics SE, I came across this differential equation: $$ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - \frac{y}{\left( 1+x^2 \right)^2}+A\frac{x}{\sqrt{1+x^2}} = 0 $$ where the ...
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On a Riemann surface, $p$ is a Weierstrass point iff the Wronskian verifies $W(0) = 0$

Let $X$ be a compact connected Riemann surface. Let $\{h_1(z),\ldots,h_g(z)\}$ be a base of the space of holomorphic forms on $X$ in a local map centered in $p\in X$. I’d like to know how to show that ...
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Exactly one homogeneous differential equation of second order to given fundamental solution

I am working on: Let $\phi_1,\phi_2$, so that $\phi_1(x)\phi_2'(x)-\phi_1'(x)\phi_2(x)\neq 0.$ for all $x\in\mathbb{R}$. Then there exists exactly on homogeneous differential equation of second order $...
john_psl1298's user avatar
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A Property of Wronskians

Let $f,g$ be smooth functions near $0$. Suppose the Wronskian determinant $$ W(f,g)(t)=f(t)g'(t)-f'(t)g(t) $$ is constantly zero. Is it true that $W(f',g')$ is also constantly zero? Note that we are ...
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If the Wronskian of two arbitrary functions is zero, are they linearly dependent only if they are not zero at some point?

Given that the functions are not the solutions of the same linear differential equation, I know that $W = 0$ doesn't mean they are linearly dependent, for example $x^3$ and $ |x|^3$ are linearly ...
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Two-variable Wronskian; regularity of coefficients

Let $f(x,y),g(x,y)$ be two real-analytic functions near a neighbourhood of $(0,0)$. Consider the following Wronskian determinant: $$ D(x,y):=\det \begin{bmatrix} f(x,y) & g(x,y)\\ \partial_x f(x,y)...
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Wronskian equivalent for difference equations?

For an $n$th order linear differential equation, if you have $n-1$ independent solutions, the Wronskian lets you find the last one. I would like to know if there is a way of finding the last solution ...
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Linear dependence of three vector functions

I am trying to determine whether the vector functions x1 = col(e^x, 2e^x, 3e^x) x2 = col(2e^2x, 4e^2x, 6e^2x) x3 = col(3e^3x, 6e^3x, 9e^3x) are linearly independent over all real numbers x. My ...
Ahdhehshdjdj's user avatar
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How do I find Wronskian for $y(t) = (c_1, tc_2, t^2c_3)e^{at}$.

If I try to use the formula it becomes very difficult to compute: $$y'(t) = ac_1e^{at} + atc_2e^{at}+ c_2e^{at} + at^2c_3e^{at} + 2tc_3e^{at}$$ and then continue with $y''$ and use the formula it ...
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Deducing relations between solutions of a linear second order ODE

I've been stuck with this problem for a couple of days: Let $u(t),v(t),w(t)$ solutions to the differential equation $y'''+y=0$, such that $u(0)=1, u'(0)=u''(0)=0$, $v(0)=v''(0)=0, v'(0)=1$ and $w(0)=...
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Is there some connection between the Wronskian determinant and Sobolev spaces, e.g. $H^1$?

Is there some connection between the Wronskian determinant and Sobolev spaces, e.g. $H^1$? I know that we often seek solutions to differential equations in Sobolev spaces because these spaces place ...
1Teaches2Learn's user avatar
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Verify that $x_1$ is a solution of Wronskian

Following from this problem on time independence of Wronskian Let $q:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. If $x_1(t)$ and $x_2(t)$ are the solutions to ODE: $\ddot{x}=q(t)x$ on $(...
variableXYZ's user avatar
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Show that Wronskian is time independent

Let $q:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. If $x_1(t)$ and $x_2(t)$ are the solutions to ODE: $\ddot{x}=q(t)x$ on $(a,b)$, show that Wronskian determinate $$ \begin{vmatrix} x_1(...
variableXYZ's user avatar
2 votes
2 answers
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Check if two functions are linearly independent.

I have to check if $f(s)=s,~g(s)=e^{ks},~k\in\mathbb{C}$ are linearly independent over $\mathbb{R}$. The wronskian is $W[s,e^{ks}]=e^{ks}(ks-1)$. Then, if we take $s=1,~k=1+0i$, it becomes $0$ and ...
Fabrizio Gambelín's user avatar
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Using Wronskian to solve nonhomegeneous ODE

I have the given ODE: $$y''+2y'+2y=e^{-x}\sin x$$ This has the homogeneous solution $y_h=C_1\cos(i-1)x+C_2\sin(-i-1)x$. The particular solution, in the form $y_p=uy_1+vy_2$, we seek the Ansatz: $y_p=...
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Finding missing 2 solutions of a 4th order ODE using Wronskian

I am trying to follow a research paper in physics where the authors end up with the following differential equation: $$ D^2(f)=0, \qquad D=\left(\frac{\rm d^2}{{\rm d}t^2} + \frac1r \frac{\rm d}{{\rm ...
squille's user avatar
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Solving 2nd Order non homogeneous differential equation using Wronskian when one solution is given

> Solve by variation of parameters $x^2 y'' + 2xy' - 6y = 5x^4$ and given that $x^2$ is a solution of homogeneous equation. Hint: Find other fundamental solution using Wronsian. My Attempt
Chamika Jayasinghe's user avatar
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Calculate the Wronskian given initial conditions

Given a differential equation $$ x y''(x) - (1 - x^2) y'(x) - (1 + x) y(x) = 0 $$ and a solution $ y_1(x) = 1 - x $, we were asked to compute the Wronskian for a second independent $ y_2(x) $ which ...
Grotto Box's user avatar
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Does the Wronskian of three or more linearly independent functions change its sign?

If $y_1, y_2$ are two Linearly independent solutions of a differential equation of order $2$ then we know that if the Wronskian is not zero then it never changes its sign (using the Abel's identity). ...
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Counter example for Wronskian Concept .

I know the result that if $y_1$ and $y_2$ are two solutions of the differential equation $$y’’+p(x)y’+Q(x)y=f(x)$$ then Wronskian $W(y_1,y_2)=ce^{\int -p(x)dx}$ of $y_1$ and $y_2$ is given by Abels ...
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Do Wronskians have the intermediate value property?

I wonder if the following is true: Conjecture: Let $I \subset \Bbb R$ be an open interval and $f, g: I \to \Bbb R$ be differentiable functions. Then the Wronskian $$ W(f,g) =\begin{vmatrix}f &g \\...
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Does there exists two differentiable functions $f, g$ on $I$ such that $W(f, g) (x) >0$ on $A$ and $W(f, g) <0$ on $I\setminus A$?

Let $I=(0, 1) $ and $A=\mathcal{C}\cap (0, 1) $ where $\mathcal{C}$ denote Cantor set. $\color{red}{Question}$ : Does there exists two differentiable functions $f, g$ on $I$ such that $W(f, g) (x) >...
Sourav Ghosh's user avatar
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Does there exists two functions $f, g\in C^1(I)$ for which $W(f, g) (x) >0$ for some $x$ and $W(f, g) (x) <0$ for some $x$?

$f, g\in C^1(I) $ where $I$ is an open interval and $f, g$ both are real valued. Let $W(f,g)(x) =\begin{vmatrix}f(x) &g(x) \\f'(x)&g'(x)\end{vmatrix}$ denote the Wronskian of $f, g$ at $x\in I$...
Sourav Ghosh's user avatar
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Understanding proof of "linear dependence of functions $f_1, f_2,...,f_n$ implies Wronskian of these functions is identically zero

The proof is shown in below picture. I am not able to understand the underlined part. How author concluded that, "linear dependence of $f_1,f_2,...,f_n$ implies that the linear system in ...
Akash Patalwanshi's user avatar
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Proof of Wronskian relation using induction

We have the following linear homogenous DE system $X' = AX, \tag 0$ I wanna prove with induction that $dW/dx = Tr(A)*W$ So for n=2 based on the above, we get, $A = \begin{bmatrix} a_{11} & a_{12} \...
DontWorry's user avatar
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Particular Solution of the ODE $y''-6y'+10y=e^{3x}$

I have to find the particular solution of the ODE $y''-6y'+10y=e^{3x}$ using, what our professor calls The Integration Formula which is: For the non-homogeneous ODE: $$p_0(x)y''+p_1(x)y'+p_2(x)y=r(x)$$...
Toniiiic's user avatar
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Can you row reduce the Wronskian with functions?

UPDATED BELOW Recently, I came across a problem asking me to exploit the Wronskian to determine the independence of a few functions. The functions were $\sin(x), \cos(x), x\sin(x)$, and $x\cos(x)$. I ...
Aaron's user avatar
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Wronskian and Abels formula and Linear Independence

Suppose I have a Linear second order Homogeneous, ode $$a_0(x)y''+a_1(x)y'+a_2(x)y=0~, x\in I$$ Now my doubts are Can I calculate Wronskian of the two solutions say $y_1$ and $y_2$ using abel's ...
Upstart's user avatar
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Linear dependence of ODE solutions

Let be $y'(x)=A(x)y(x)$ an ODE and $y_1(x),y_2(x)\cdots,y_n(x)$ some solutions. Let be $Y(x)$ the Wronskian of the those solutions. We have proven the statement that if we find an $x_0$ such that rank$...
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$\frac{d\theta}{dt} = \frac{W(x_{1},x_{2})}{|r|^2}$, where $r:= x_{1} i + x_{2} j$ and $\theta$ is the angle between $r$ and $i$,

The title says it all. My biggest problem is: I can't see any part of the Wronskian within the angle; actually I'm having problems to find a angle function. Let $r:= x_{1} i + x_{2} j$ be a curve and $...
big_GolfUniformIndia's user avatar
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Construction of ODE using Wronskian

I have seen some construction of a homogeneous second order linear ODE using the Wronskian $W(t)=W[y_1,y_2](t)$ of the linearly independent twice differentiable functions $y_1$ and $y_2$. The ODE is ...
Riaz's user avatar
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Difficulty Understanding a Particular Usage of Abel's Theorem in the Wronskian

suppose I have some equation: $$ \frac{dy^2}{dx^2} + (x^2-3x)y' + 12y = 0 $$ (The above equation is just for example) Using Abel's Theorem for the Wronskian: $$ W(y_1, y_2)(x) = ce^{-\int p(x)~dx} $$...
Squarepeg's user avatar
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Wronskian for non-homogeneous second order linear ODE

Q. Consider the ODE: $u''(t)+P(t)u'(t)+Q(t)u(t)=R(t),\ t\in[0,\ 1]$. There exists continuous functions $P,Q$ and $R$ defined on $[0,\ 1]$ and two solutions $u_1$ and $u_2$ of this ODE such that $W(t)=...
Riaz's user avatar
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Assuming $y_1$ and $y_2$ are two solutions of $y'' +p(x)y' +q(x)y=0$, find a first order constant coefficient DE satisfied by Wronskian, and solve it

$$y'' +p(x)y' +q(x)y=0$$ Assuming $y_1$ and $y_2$ are two solutions of the given DE, find a first order constant coefficient DE satisfied by their Wronskian, and solve it. I don't really understand ...
Ahsan Yousaf's user avatar
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2 answers
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Wronskian | Ordinary Differential Equation

I was solving a question where I was asked to prove that if $p(x) = 0$ for $L(y) = y'' + p(x)y' + q(x)y = 0$ then the wronskian of the two independent solutions is a constant. I was able to prove that ...
mathfanatic's user avatar
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1 answer
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Find $p(t)$ for the ODE given that the Wronskian is a non-zero constant.

Consider an ODE of the form $$\frac{d^2x}{dt^2}+p(t)\frac{dx}{dt}+q(t)x=0$$ Suppose that we have two solutions $x_1(t)$ and $x_2(t)$ to this ODE and their Wronskian is a non-zero constant. That is, $$...
James Anderson's user avatar
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If $x_1$ and $x_2$ are distinct solutions of the differential equation $\ddot{x}+a_1(t)\dot{x}+a_2(t)=0$, show the solutions are linearly dependent.

We are given that $a_1$ and $a_2$ are continuous functions that have a maximum or a minimum at the same $t$, for some $t\in I$. I tried using Abel's Identity with the Wronskian, showing that $$W'(x_1,...
Joe Miller's user avatar
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Wronskian of vector

$\vec Y'=AY, A_{2\times2}$ (constants over $\mathbb{R}$). $\vec Y^{(1)}(x)=\begin{pmatrix}e^{2x}\\ -e^{2x}\end{pmatrix}, W[\vec Y^{(1)},\vec Y^{(2)}](2)=e^3,W[\vec Y^{(1)},\vec Y^{(2)}](1)=e^2$. ($W :=...
Algo's user avatar
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Wronskian Differential Equation.

$3y''+(6/x)y'+3e^xy = 0$ and two $y_1$, $y_2$ are two partial solutions of such that $W(y_1, y_2) \ne 0$. (where $W(y_1, y_2) = W(x)$ is the Wronskian of $y_1$, $y_2$). If it is known that $W(1) = 2$, ...
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