Wronskian Definition and 81 Threads

Hello, I understand that if we have three functions f, g, and h, they are linearly independent <=> the only c1, c2, and c3 that satisfy (c1)f+(c2)g+(c3)h=0 are c1=c2=c3=0. In order to solve for these c1, c2, and c3, we want three equations in the three unknowns. To do this we can...

Edit: I think I may have posted this in the wrong section, sorry about that. Note that this isn't a homework problem though, I"m not enrolled in this class, I was just reading over some of this stuff and trying some problems since I"m majoring in physics. I have a textbook "discussion" problem...

Hi I am trying to do this problem. Verify that \( y_1=x^3 \) and \(y_2=|x|^3 \) are linearly independent solutions of the diff. equation \( x^2y''-4xy'+6y=0\) on the interval \((-\infty,\infty) \). Show that \( W(y_1,y_2)=0 \) for every real number x. I could actually show the above by...

Homework Statement y1(t) and y2(t), 2 solutions of the equation: y'' +ay'+by=0, with a,b εℝ - {0} a) Determine: d/dt w(y1,y2) where w(y1,y2) is the wronskian of y1(t) and y2(t) b) Deduce that if (y1(0),y1'(0)^T and (y2(0), y2'(0))^T are 2 linearly independent vectors. Then y1(t)...

Homework Statement Determine which of the following pairs of functions are linearly independent. (a) f(t)=3t,g(t)=|t| (b) f(x)=x^{2},g(x)=4|x|^{2} Homework Equations the Wronskian is defined as, W=Det{{f(u),g(u)},{f'(u),g'(u)}} if {f(u),g(u)} are linearly dependent, W=0...

Homework Statement Given the two functions: f(t) = t g(t) = |t| Use the Wronskian to determine if the two functions are dependent or independent. 2. The attempt at a solution I have already found the correct answer to this, which is that it is independent but I have some questions as...

I have a Wronskian Question? If the Wronskian W of f and g is t^2*e^t and if f(t)=t, find g(t). I have tried setting up this problem: tg'-t'g = t^2*e^t tg'-g = t^2*e^t Setting up the integrating factor, µ= e^∫-1 --> µ= e^-t (e^-t)t*g' - (e^-t)*g = (e^-t)(t^2*e^t) so preferably I...

Hi, I just want to clarify something written in my textbook - a contradiction of sorts. My book says, if i have two functions, Y1 Y2, and their wronskian is 0 at any point on the interval I, the functions are dependant functions. However, while doing a problem, I found the wronskian to...

Homework Statement Let v_1,v_2 be any two solutions of the differential equation y''+ay'+by=0 such that \frac {v_2}{v_1} is not constant, and let f(x) be any solution of the differential equation as well. Use the properties of the Wronskian to prove that constants c_1,c_2 exist such that: c_1...

Homework Statement Given a second order differential equation: y'' + P(x)y' + Q(x)y = 0 If y1(x) and y2(x) are linearly independent solutions of the DE, what form does Abel's Equation give for W(y1(x), y2(x))? If we assume that one solution y1(x) is known, what first order DE results from a...

For y1 = t2 and y2 = t|t| (y2'' is not defined at t = 0), the Wronskian is 0 for all t over the interval [-1,1]. However, the two functions are not linearly dependent over this interval in the sense that one is not a unique multiple of the other. Does this imply that the Wronskian tells linear...

Hello I'm trying to solve the following problem: given the scalar ODE x''+q(t)x=0 with a continuous function q. x(t) and y(t) are two solution of the ODE and the wronskian is: W(t):=x(t)y'(t)-x'(t)y(t). x(t) and y(t) are linear independent if W(t)\neq 0. I want to show that W(t) is...

I'm reading Ince on ODEs, and I'm in the section (in Chapter 5) where he talks about the Wronskian. There are quite a few things here that I don't quite understand or follow. I'm not going to get into all the details, but briefly, suppose we have the Wronskian of k functions: W =...

Homework Statement Hi I seem to remember that if you have a homogenous ODE y'' + p(t)y' + q(t)y = 0 which have the solutions y1 and y2. Where we are told that y1(t) \neq 0 then y1 and y2 are linear independent. I found the simular claim on sosmath.com but are they simply...

Hi Homework Statement I have differential equation y'' + p(x)\cdot x' + q(x)y = 0 which have two solutions y_1(x) and y_2(x) where y_1(x) \neq 0 show that y_2(t) = y_1(t)\int_{t_0}^{t} \frac{1}{y_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds is also a solution. Homework Equations I...

Homework Statement Hey Everyone, Here is a problem from my book that has my confused. I really don't understand what it wants me to do so if anyone could give me a few hints it would be greatly appreciated. I am doing problem 34, but I included 33 since it wanted to follow the same method...

Homework Statement Okay so the question is to show that these 2 functions are linearly dependent. ie. they are not both solutions to the same 2nd order, linear, homogeneous differential equation for non zero choices of, say M, B and V Homework Equations f(x) = sin(Mx) g(x) = Bx + V...

Homework Statement Find the Wronskian W(t)=W(y1,y2) where I have found y1=1 and y2=(2/9)-(2/9)e^(-9t/2) The Attempt at a Solution I am not sure how to do the Wronskian. We haven't talked about at all in class and I am not even sure what exactly it does. Any help would be greatly...

Is there any exception where I can't use wronskian rule to see if given functions are linearly independent or dependent? Thanks...

Homework Statement Given that Φ2 = Φ1 * ∫ e^(-∫a(x)dx)) / (Φ1)^2 dx and Φ1 = cos(ln(x)), a = 1/x, solve for Φ2. Homework Equations The Attempt at a Solution Φ2 = cos(ln(x)) * ∫ e^(-∫1/x dx)) / cos^(2)(ln(x)) dx = cos(ln(x)) * ∫ e^(-ln(x)) / cos^(2)(ln(x)) dx = cos(ln(x)) * - ∫ x /...

Homework Statement take the wronskian of [cos(theta)]^2 and 1+cos(2theta) Homework Equations The Attempt at a Solution so I set up the determinant [cos(theta)]^2 1+cos(2theta) as my y1 and y2 respectively and -2cos(theta)sin(theta) and -sin(2theta) as my y1 and y2 prime...

I haven' been able to find good explanations of either of these: Part 1: Jordan Normal Form: Is this it? An n*n matrix A is not diagonizable (ie. A=PDP^-1) because it has linearly dependent eigenvectors (no. of eigenvectors is less than n). However, it can be expressed in a similar form...

Homework Statement f1 = 0 , f2 = x , f3 = e^x I am supposed to find out if these are linearly independent or dependent. Just by looking at it, I can't see a way to write one of the functions as a combination of the other two with constant multiples, so to make sure that it is linearly...

In general, the question is how do you take the derivative of the determinant of a matrix of functions, but more specifically how does one do this for a Wronskian? I've read a remark that seemed to say that the derivative for an nth order Wronskian is the determinant of a sum of n matrices...

"For the Wronskian, W, Show W(x,fg,fh)=([f(x)]^2)W(g,h)" How is this done? I know how to use the Wronskian when there's a system of equations, something like y(x) = cosx, y(x)=sinx, y(x)=x, etc. But I'm really clueless about how to proceed here.

Homework Statement Hi, could someone please confirm my results. I just put my answers because the procedure is so long. let me know if you get the same results. 1) Wronskian(e^x, e^-x, sinh(x)) = 0 2) Wronskian(cos(ln(x)), sin(ln(x)) = 1/x * [cos^2(ln(x)) + sin^2(ln(x))] = 1/x thanks in...

What's the wronskian of x^2 and x^-2? I've found a basis of solutions to a non-homogeneous 2nd order ODE and want to find a particuler solution using variation of parameters.

y''' + 25y' = csc(5x) i got the y (complimentary) = C1 + C2cos5x + C3sin5x. I'm just having minor difficulties getting the y (particular).

When dealing with Abel's formula for the wronskian of a second order ODE: W(R)=Ce^{-\int p_1(R)dR} and assuming that you don't know the homogeneous solutions but you know their asymptotic behavior at infinity and at the origin, how is the constant C calculated? Thanks.

Hello out there. I'm working on a proof by induction of the Wronskian and need a little boost to get going. So, here goes: If y_1,...,y_n \in C^n[a,b], then their Wronskian is...

w[f,g](t)= t^2\exp{t}\\f(t)=t Thats what i get, the problem is to find g(t) So, i start; f'(t)=1 w[f,g](t)= t^2\exp{t}=f(t)g'(t)-f'(t)g(t)\\t^2\exp{t}=tg'(t)-g(t) divide by t, t\exp{t}=g'(t)-\frac{g(t)}{t} its a 1st order linear eq. I solve for the integrating factor and...