Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
34 views

What is a Wronskian and why is it useful here?

A particle in the region $z$ is described by $\psi(z, p)=A(p) f(z, p)$. Where $A(p)$ is independent of $z$ and $f(z,p)$ is the solution to: \begin{equation} \left[\frac{\partial^2}{\partial z^2}+\frac{...
Tomi's user avatar
  • 160
0 votes
0 answers
26 views

Are $\tau$ and $t$ given as boundary/initial conditions? If not, how to find both?

Recall from the Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE page that if we have a linear homogeneous $n^{\text {th }}$ order ODE $y^{(n)}+a_{n-1}(t) y^{(n-1)}+\ldots+a_1(t) y^{...
user avatar
0 votes
1 answer
61 views

Where does this expression for Green's function come from?

In the context of $2$nd order ODEs, I found in some solution sheet that they computed the Green's function using the following expression $$G(x;x')=\dfrac{y_2(x)y_1(x')-y_1(x)y_2(x')}{\overline{\...
Conreu's user avatar
  • 2,523
3 votes
0 answers
53 views

Is this a valid formula for the Wronskian?

I was messing around with the Wronskian of two functions $y_1(x)$ and $y_2(x)$, which is defined by: $$ W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1y_2'-y_2y_1^{\...
cherrytree's user avatar
2 votes
0 answers
72 views

Differential equation by the parameter variation method

I need help to solve this equation by the method of parameters variation (Wronskian): $$4\,y''-3\,y'=x\,e^{\frac{3}{4}\,x}$$ I know the solution of the differential equation is: $$y=\dfrac{{x}^{2}\,{e}...
Bass's user avatar
  • 51
2 votes
1 answer
98 views

Introduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there won't be a term with $z'$ in the new equation.

Introduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there won't be a term with $z'$ in the new equation. Find all solutions of such equations and also explore the possibility when $z'=...
user avatar
1 vote
0 answers
25 views

Why only test Wronskian at single time t rather than over the entire interval the solution exists

When determining linear independence of solutions to a linear ODE, why do we only need to test the Wronskian at a single time t, rather than over the entire interval in which the solution exists?
SayMyNameHeisenberg's user avatar
0 votes
2 answers
73 views

How can I solve a Bessel equation with Reduction of order?

If $y_{1}(x) = \frac{\sin(x)}{\sqrt(x)}$ is one solution of the differential equation $$x^2y'' +xy' + (x^2-\frac{1}{4})y = 0$$ find the second solution $y_{2}(x)$. My effort using Wronskian The ...
Homer Jay Simpson's user avatar
2 votes
1 answer
72 views

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 ...
AntidusPig's user avatar
2 votes
1 answer
121 views

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 ...
Gajjze's user avatar
  • 386
0 votes
1 answer
98 views

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, $x$, ${x^2}$....
klasser's user avatar
  • 33
1 vote
1 answer
82 views

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 ...
oleout's user avatar
  • 1,180
1 vote
1 answer
75 views

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 ...
user5587's user avatar
0 votes
1 answer
15 views

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 $...
number8's user avatar
  • 577
0 votes
0 answers
82 views

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
  • 1,277
1 vote
0 answers
32 views

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)=...
math student's user avatar
  • 1,277
0 votes
0 answers
118 views

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 ...
calculatormathematical's user avatar
0 votes
0 answers
27 views

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 $...
James H's user avatar
  • 304
1 vote
1 answer
56 views

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
1 vote
2 answers
89 views

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 ...
Jonathan Huang's user avatar
0 votes
1 answer
112 views

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 ...
dahemar's user avatar
  • 1,788
4 votes
1 answer
55 views

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
1 vote
1 answer
100 views

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 ...
Tongou Yang's user avatar
  • 2,015
1 vote
1 answer
198 views

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 ...
EB97's user avatar
  • 164
0 votes
0 answers
46 views

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)...
Tongou Yang's user avatar
  • 2,015
2 votes
0 answers
86 views

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 ...
Zoe Allen's user avatar
  • 5,548
-1 votes
1 answer
48 views

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
0 votes
1 answer
57 views

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 ...
Ashh3720's user avatar
0 votes
1 answer
25 views

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)=...
madame p's user avatar
  • 147
2 votes
0 answers
55 views

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
0 votes
1 answer
34 views

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
  • 1,073
1 vote
1 answer
68 views

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
  • 1,073
2 votes
2 answers
343 views

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 G's user avatar
  • 2,107
2 votes
2 answers
102 views

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=...
Superunknown's user avatar
  • 2,973
2 votes
1 answer
59 views

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
  • 123
2 votes
1 answer
85 views

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
1 vote
1 answer
141 views

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
3 votes
1 answer
527 views

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). ...
ThirstForMaths's user avatar
1 vote
0 answers
86 views

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 ...
neelkanth's user avatar
  • 6,090
13 votes
1 answer
393 views

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 \\...
Martin R's user avatar
  • 117k
2 votes
1 answer
203 views

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) >...
Ussesjskskns's user avatar
2 votes
1 answer
200 views

Does there exist 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 $x\in I$...
Ussesjskskns's user avatar
0 votes
0 answers
130 views

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
2 votes
1 answer
107 views

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
  • 131
0 votes
3 answers
212 views

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
  • 205
2 votes
2 answers
360 views

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
  • 23
2 votes
0 answers
53 views

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
  • 2,632
0 votes
1 answer
82 views

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$...
Philipp's user avatar
  • 4,564
0 votes
1 answer
21 views

$\frac{d\theta}{dt} = \frac{W(x_{1},x_{2})}{|r|^2}$, 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
0 votes
1 answer
76 views

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
  • 2,184

1
2 3 4 5