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.
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-...
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 Equations
The...
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= (y2`y1)-(y2y1`)
v1 = integral of ( g(t) (y1) ) / W
v2 = integral of ( g(t) (y2) ) / W
The Attempt at a Solution
[/B]
yc= c1...
Homework Statement
x²y''+xy'+(x²-0,25)y=0
y1= x^-1/2*sin x
Homework 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...
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...
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...
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...
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...
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
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...
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...
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...