- #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?