# wronskian

1. ### 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-...
2. ### 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 Equations The...
3. ### 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...
4. ### Abel's Equation and Wronskian for find out y2

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...
5. ### 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...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### Fundamental matrix vs Wronskian

I have just learnt 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
10. ### 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...
11. ### 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...
12. ### Integrating sine^x

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...
13. ### Help with Wronskian Equation

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...