# Wronskian Problem

Printable View

• July 26th 2009, 11:26 AM
diddledabble
Wronskian Problem
I don't understand how to do these type of problems. Verify that the given function forms a fundamental solution set and then find the general solution using Wronskian.

$y'''-y''+4y'-4y=0$
Given ${e^{x},cos2x,sin2x}$
• August 1st 2009, 12:57 PM
Media_Man
Wronskian - Wikipedia, the free encyclopedia

$W(e^x,\sin(2x),\cos(2x))(x)=\left|\begin{array}{cc c}e^x & \sin(2x) & \cos(2x)
\\e^x & -2\cos(2x) & 2\sin(2x) \\e^x & -4\sin(2x) & \ -4\cos(2x)\end{array}\right|=10e^x$

Verifying that the Wronskian returns a nonzero determinant, these three functions represent linearly independent solutions. Thus the general solution is a linear combination of them:

$y(x)=Ae^x+B\sin(2x)+C\cos(2x)$