- #1

c0der

- 54

- 0

## Homework Statement

Solve:

[ d^2y1/dx^2 ] = [ a -a ] [ y1 ]

[ d^2y2/dx^2 ] [ -a a ] [ y2 ]

A = [ a -a ]

[ -a a ]

## Homework Equations

Everything required is in (1) above

## The Attempt at a Solution

Reduce to 1st order system

M = [ 0 I ]

[ A 0 ]

Hence, M =

[ 0 0 1 0 ]

[ 0 0 0 1 ]

[ a -a 0 0 ]

[ -a a 0 0 ]

The eigenvalues of M are 0, 0, √2a and -√2a

The eigenvectors are [ 1 1 ], [ 1 1 ], [ 1 -1] and [ 1 -1]

Hence the general solution is (for y only):

y(x) = A*e^(0x)*[ 1 1 ]T + B*e^(0x)*[ 1 1 ]T + C*e^(√2a*x) [ 1 -1 ]T +

D*e^(-√2a*x) [ 1 -1 ]T

So:

y1 = A + B + C*cosh(√(2a)*x) + D*sinh(√(2a)*x)

y2 = A + B - C*cosh(√(2a)*x) - D*sinh(√(2a)*x)

Is this correct?