What is Wronskian: Definition and 87 Discussions

In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by Józef Hoene-Wroński (1812) and named by Thomas Muir (1882, Chapter XVIII). It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

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1. Linearly independent functions with identically zero Wronskian

My question will be about item (c). Part (a) Note that for ##x\geq 0## we have ##f(x)=g(x)##. For ##x<0## we have ##f(x)=-g(x)##. Since ##f## is a constant times ##g## then one column of the matrix in the Wronskian is a constant times the other column. Thus, the Wronskian is zero, Note that...
2. Understanding Wronskian of solutions to homoeneous 2nd order linear DE

The Wronskian of these two solutions is also a function of ##x##. $$W=y_1y_2'-y_1'y_2$$ $$W'=y_1y_2''+y_1'y_2'-y_1'y_2'-y_1''y_2$$ $$=y_1y_2''-y_1''y_2$$ The two solutions satisfy $$y_1''+Py_1'+Qy_1=0$$ $$y_2''+Py_2'+Qy_2=0$$ Multiply the first by ##y_2## and the second by ##y_1## and...
3. I Determining linear indepedence/dependence of a set of functions

Looking at the wronskian applications- came across this; Okay, i noted that one can also have this approach(just differentiate directly). Sharing just incase one has more insight. ##-18c \sin 2x -4k\cos x \sin x - 4k\sin x\cos x =0## ##-18c\sin 2x-2k\sin2x-2k\sin 2x=0## ##-18c\sin 2x =...
4. MHB -m30 - 2nd order linear homogeneous ODE solve using Wronskian

2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...
5. Find the Wronskian of solutions y1 and y2 of the equation

Homework Statement [/B] Homework Equations ##W(x) = C exp (∫ P(x) dx)## ##y''+P(x)y'+g(x)=0## The Attempt at a Solution Divide the origin equation by (x) to get the standard form of the homogeneous equation : ##y'' +\frac{3y'}{x} - \frac{3y}{x}## ##P(x) = \frac{3}{x}## ## exp (- ∫...
6. Use Abel's identity to find the Wronskian

Homework Statement Homework Equations The Attempt at a Solution [/B] Is my answer correct ?
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A Numerics: Wronskian and linear independence

Hi PF! I'm solving a differential eigen-value problem in weak form, so I have trial functions. If the Wronskian of trial functions is small but not zero, is linear independence an issue? I have analytic trial functions but am numerically integrating.
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A Variation of parameters, Green's functions, Wronskian

Hi PF! I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier...
9. MHB Wronskian of Airy functions.

I am trying to show that that the Airy functions defined below satisfy: $W[Ai(x),Bi(x)]=1/\pi$. $$Ai(x)=\frac{1}{\pi} \int_0^\infty \cos(t^3/3+xt)dt$$ $$Bi(x)=\frac{1}{\pi}\int_0^\infty \bigg[ \exp(-t^3/3+xt)+\sin(t^3/3+xt)\bigg]dt$$ I tried to compute it directly but I got stuck, here's the...
10. I Constructing a 2nd order homogenous DE given fundamental solution

Homework Statement Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)} Homework Equations y''+p(x)y'+q(x)=0 det W(y1,y2) =Ce-∫p(x)dx The Attempt at a Solution I took the determinant of the matrix to get e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...
11. I Difference between Wronskian and other methods

I am confused about determining when two or more functions are linearly independent. My textbook notes that the Wronskian can do this, but then it also mentions the definition of linear Independence, that the linear combination ##c_1 f_1 + c_2 f_2 + ... + c_n f_n = 0## only has the trivial...
12. Wronskian Property: Linearly Dependent Functions

Homework Statement If two differentiable functions are linearly dependent on the interval I, then their Wronskian is identically zero on I. Homework EquationsThe Attempt at a Solution We start with the definition of linearly dependence for two functions. ##y_1 = Cy_2## ##\displaystyle...
13. A Wronskian- variation of Params Problem

Homework Statement y''-4y'+4y=(12e^2x)/(x^4) I am trying to solve this differential equation. I know you would use the variation of parameters method, and I am trouble with the wronskian. My solution manual doesn't actually use a wronskian so I can't verify my work Homework EquationsThe...
14. Solve for the solution of the differential equation

Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x - x = (e^t) + t Homework Equations [/B] W= (y2y1)-(y2y1) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...
15. Abel's Equation and Wronskian for find out y2

Homework Statement x²y''+xy'+(x²-0,25)y=0 y1= x^-1/2*sin xHomework Equations Abel's equation: W= c.e^-(integrate (p(t))The Attempt at a Solution My Wronskian gave me a first order ODE that I really don't know solve. x^-1/2*sinx y' + (1/2 x^-3/2 sin x- x^-1/2cosx) y2 I don't solved the Abel's...
16. Wronskian and differential equations

Homework Statement The problems are in the uploaded file. 18) satisfies the differential equation y''+p(t)*y'+q(t)*y=0 p(t) and q(t) are continuous Homework Equations Wronskian of y1 and y2 The Attempt at a Solution 18) I don't really get this one 19) Solved most but at the end, where I...
17. Wronskian and two solutions being independent

In the uploaded file, question 11 says that in b) the solutions y1 and y2 are linearly independent but the Wronskian equals 0. I think it said that they are independent because it's not a fixed constant times the other solution (-1 for -1<=t<=0 and 1 for 0<=t<=1) but it clearly says in the...
18. MHB Proving Wronskian of $(\star)$ System using Hint

Hello! (Wave) Prove that the determinant of the following system $(\star)$ is the Wronskian.$$(\star) \begin{pmatrix} y_1(s) & -y_2(s)\\ -y_1'(s) & y_2'(s) \end{pmatrix} \begin{pmatrix} c_1(s)\\ c_2(s) \end{pmatrix}=\begin{pmatrix} 0\\ \frac{1}{p(s)} \end{pmatrix}$$Hint: Write the equation...
19. Need help interpreting the Wronskian

I'm given bases for a solution space \left \{ x,xe^x,x^2e^x \right \}. Clearly these form a basis (are linearly independent). But, unless I've made a mistake, doing the Wronskian on this yields W(x) = x^3e^x. Isn't this Wronskian equal to zero at x = 0? Isn't that a problem for...
20. Using Abel's Theorem, find the Wronskian

Using Abel's thrm, find the wronskian between 2 soltions of the second order, linear ODE: x''+1/sqrt(t^3)x'+t^2x=0 t>0 I think I got the interal of 1/sqrt(t^3) to be 2t/sqrt(t^3) but this is very different to the other examples I've done where a ln is formed to cancel out the e in the formula...
21. Fundamental matrix vs Wronskian

I have just learned the first order system of ODE, i found that the Wronskian in second order ODE is |y1 y2 ; y1' y2'| but in first order system of ODE is the Wronskian is W(two solution), i wonder which ones is the general form? thank you very much
22. Rewriting Third Column: Wronskian Matrix Homework Guide

Homework Statement Page 133 Homework Equations n/a The Attempt at a Solution What is the process for rewriting the third column? 2x-3 and be rewritten as 2x, and 2-3cosx can be rewritten as 2. I don't get this.
23. Construct a second order ODE given the solutions?

Homework Statement I've been stuck on this problem for three days now, and I have no clue how to solve it. Construct a linear differential equation of order 2, for which { y_1(x) = sin(x), y_2(x) = xsin(x)} is a set of fundamental solutions on I = (0,\pi) . Homework Equations Wronskian for...
24. If the Wronskian equals 0, is it always 0?

If the Wronskian of a set of equations equals 0 over a particular interval in the functions' domain, is it possible for it be non-zero under another interval? Are there any particular proofs for or against this?
25. Linear Dependence/Independence and Wronskian

So I know there are a few threads and many websites on this, but I am not finding what I am looking for. To determine whether a set of functions are linearly dependent or independent I understand that the Wronskian can be used, but many example problems state that "clearly by inspection" some...
26. MHB Use Wronskian to show only 2 indepndent solutions of 2nd order ODE

Given standard ODE $y'' + P(x)y' + Q(x)y=0$, use wronskian to show it cannot have 3 independent sltns. Assume a 3rd solution and show W vanishes for all x. so 1st row of W = {${y}_{1}, {y}_{2},{y}_{3}$}, 2nd row is 1st derivatives, 3rd row is 2nd derivatives. I can find the determinate...
27. Wronskian to determine L.D

Homework Statement Hello, I was just looking for a quick tip: If I have three distinct solutions to a second order linear homogeneous d.e, how would I show that the wronskian of (y1,y2,y3)(x)=0? I know how to show the wronskian is not zero for a linearly independent set, but I'm confused...
28. Find Wronskian of {e^(x)*cos(sqrt(x)), e^(x)*sin(sqrt(x))} Homework

Homework Statement Find the Wronskian of {e^(x)*cos(sqrt(x)), e^(x)*sin(sqrt(x))}. Homework Equations W(f, g)=fg'-gf' The Attempt at a Solution W(f, g)=(e^(x)*cos(sqrt(x)))(e^(x)*cos(sqrt(x))*1/(2x^1/2))-(e^(x)*sin(sqrt(x)))(-e^(x)*sin(sqrt(x))*1/(2x^1/2)+e^(x)*cos(sqrt(x))) After simplifying...
29. Calculators How to find the Wronskian on Ti-89?

I'm doing a problem about finding the Wronskian but I want to check my answer if it's right or not. How do I find the Wronskian of that problem on Ti-89 Titanium?
30. MHB Understanding the Wronskian Determinant

Hi guys! How's it going? OK so I'm like totally stuck and I have no idea how to do this.
31. MHB Compute the Wronskian & Simplify

Compute the Wronskian and simplify. So the first part was easy a) $y_1 = t^2 + 1$ , $y_2 = 3t^2 + k$ =$6t-2kt$ b) for what values of $k$ are the functions linearly independant so would I just solve for $k$? I'm confused What exactly does linearly independant mean?
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What is the purpose of the Wronskian in solving second order linear ODEs?

Hi pf! If ##y_1## and ##y_2## are homogenous solutions to a (not necessarily homogenous) second order linear ODE, we define the Wronskian as $$\begin{vmatrix} y_1 y_2\\ y'_1 y'_2 \end{vmatrix}$$. This derivation seems to stem from the pair of equations involving ##y_1## and ##y_2## satisfying...
33. MHB Linearly independence using Wronskian?

Hi, so I am given this problem: Using the Wronskian, show that 1, x, x^2,..., x^(n-1) for n>1 are linearly independent. The wronskian is not zero for at least one value in the interval so it is linearly independent, I just do not know how to show it properly.Thank you! :D
34. What is the connection between the Wronskian and linear independence?

I was just curious and had a question: why does the Wronskian indicate linear independence if ## W ≠ 0 ## but is linearly dependent if ## W = 0 ##? Is there a proof to help understand the exact operations of the Wronskian and why it conveys these properties based on these results alone? Thank you!
35. How does the Wronskian determine uniqueness of solution?

Given an nth order DE, how (intuitively and/or mathematically) does computing the Wronskian to be nonzero for at least one point in the defined interval for the solution to the DE ensure the solution is unique and also a fundamental set of solutions? Also, is it true that if W = 0, it is 0 for...
36. Solve sine^x Variation of Parameters: y"+3y'+2y

Homework Statement Solve by variation of parameters: y" + 3y' + 2y = sinex Homework Equations Finding the complimentary yields: yc = c1e-x + c2e-2x The Attempt at a Solution I set up the Wronskians and got: μ1 = ∫e-2xsin(ex)dx μ2 = -∫e-xsin(ex)dx The problem is that I have no idea how to...
37. Wronskian Equation for y1 and y2 with Initial Conditions

Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t)Homework Equations W(y1, y2) = |y1 y2...
38. Wronskian of the confluent hypergeometric functions

According to [Erdely A,1953; Higher Transcendental Functions, Vol I, Ch. VI.] the confluent hypergeometric equation \frac{d^2}{d x^2} y + \left(c - x \right) \frac{d}{d x} y - a y = 0 has got eight solutions, which are the followings: y_1=M[a,c,x] y_2=x^{1-c}M[a-c+1,2-c,x]...
39. Where Can I Find a Proof for the Wronskian Formula?

Can anyone point to a proof for the Wronskian formula: W[v,v*] = v'v* - vv*' = 2iIM(v', v*) Thanks
40. Wronskian vs. Determinant in Determining Linear Independence?

It seems to me that if a row is able to be zeroed out through Gaussian reduction that the determinate of that matrix would equal zero. Therefore, we know that at least one of equations/vectors that constructed the matrix was formed from the other two rows. That is -- that equation is dependent...
41. Finding Wronskian of Diff. Eq. with Abel's Formula: Help Needed

1. Use Abel's Formula to find the Wronskian of two solutions of the given differential equation without solving the equation. x2y" - x(x+2)y' + (t + 2)y = 0 2. Abel's Formula W(y1, y2)(x) = ce-∫p(x)dx3. I put it in the form of y" + p(x)y' + q(x)y = 0 to find my p(x) to use for Abel's...
42. MHB Using the Wronskian to Solve Differential Equations with Non-Constant Solutions

Hey! :) I need some help at the following exercise: Let v_{1}, v_{2} solutions of the differential equation y''+ay'+by=0 (where a and b real constants)so that \frac{v_{1}}{v_{2}} is not constant.If y=f(x) any solution of the differential equation ,use the identities of the Wronskian to show...
43. Calculating Wronskian for u and v: f+3g and f-g

If the Wronskian of f and g is tcos(t)-sin(t), and if u=f+3g, v=f-g, find the Wronskian of u and v. W=fg'-gf' (f+3g)(f-g)'-(f+3g)'(f-g) What's next?
44. Wronskian to determine lin.ind. of solutions to a system of ODEs

Homework Statement In my book, I'm given that ##\vec{x}_1=\left(\begin{matrix}t^2\\t\end{matrix}\right), \vec{x}_2=\left(\begin{matrix}0\\1+t\end{matrix}\right), \vec{x}_3=\left(\begin{matrix}-t^2\\1\end{matrix}\right)## are solutions. My textbook presents an algebraic way to show that the...
45. Is the Sign of the Wronskian Arbitrary in Differential Equations?

On my DE test, I was asked to determine if two solutions to a DE are fundamental solutions. So I confirmed they were both solutions, and took the Wronskian, which was nonzero. I got points marked off, and he put a minus sign in front of my wronskian result. Isn't the sign of the...
46. D.E. Third-order Wronskian formulas

I can't find a reference for what the wronskian formulas are when dealing with a 3rd order D.E. I know that: W= W[y1,y2,y3] W1= 1*W[y2,y3] W2= -1* W[y1,y3] W3= ?
47. How to Determine Linear Dependence Using Wronskian: A Quick Guide

urgent Diff. Eqs. Wronskian question Homework Statement See attached image- it's a lot easier. Homework Equations We know that when the wronskina = 0, it is linearly dependent on most points, and if it is not equal to 0, then the solutions form a fundamental set of solutions because...
48. D.E. Wronskian Method: Stuck trying to show L.I. and L.D. intervals

I need to show the intervals where the wronskian is linearly independent and linearly dependent.. I don't know how to do that.. Here's what I have:
49. MHB RF's question at Yahoo Answers (linear independence, Wronskian).

Here is the question: Here is a link to the question: Differential Equations...Linear independence question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
50. Checking Linear Independence. Using Wronskian vs. Using Definition

Homework Statement Is the set $$\{cos(x), cos(2x)\}$$ linearly independent?Homework Equations Definition: Linear Independence A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero such that a linear combination of the functions in the set are equal to...