Eigenvalue Method: Solving 2nd Order ODEs

[kahless2005]

Given:Second order ODE: x" + 2x' + 3x = 0
Find:
a) Write equation as first order ODE
b) Apply eigenvalue method to find general soln

Solution:

Part a, is easy
a) y' = -2y - 3x

now, how do I do part b? Do I solve it as a [1x2] matrix?

 

[eigenglue]

I don't think you have part a quite correct. It believe should be a matrix equation, something like z' = Az, where z is vector and A is a 2x2 matrix. You would then use the eigenvalue method on the 2x2 matrix.

 

[pocoman]

Your solution of part a is wrong. I think you should define the vector
$$u=\left(\begin{array}{cc}x'\\x\end{array}\right)$$
so the derivative of u:
$$u'=\left(\begin{array}{cc}x''\\x'\end{array}\right)$$
By substituting x''=- 2x' - 3x into u'=(x'';x'), you get:
$$u'=\left(\begin{array}{cc}- 2x' - 3x\\x'\end{array}\right)$$
and will easily find the solution, something like u' = Au + B as eigenglue said.