Question: Characteristic relation for linear ODE system?

hi.for linear coupling equation

dsys3 := {-72.49829200*(diff(f1(x), x, x))+0.8377580411e-2*(diff(f2(x), x))-8.873545400*10^9*(diff(f3(x), x))+2.114533515*10^18*f1(x), -878.8477313*(diff(f2(x), x, x))+1.590065471*10^20*f2(x)-7.353421206*10^(-26)*(diff(f3(x), x, x))+4.891459762*10^10*f3(x), 4.027667395*10^(-20)*(diff(f3(x), x, x, x, x))-0.6274394007e-2*(diff(f3(x), x, x))+8.873545401*10^9*(diff(f1(x), x))-7.352113720*10^(-26)*(diff(f2(x), x, x))+4.904509456*10^10*f2(x)+1.208381068*10^19*f3(x)-2.499990383*10^26*omega*(diff(f3(x), x, x))}

by assuming and to expand these functions(f1-f2-f3) in polynomial form(e.g. Chebyshev, power polynomials, Legendre and etc).:

f1(x):=(∑)H[i] *(e)^(lambda[i] *x); f2(x):=(∑)alpha[i]*H[i] *(e)^(lambda[i] *x);f3(x):=(∑)GAMMA[i]*H[i] *(e)^(lambda[i] *x)

how i detemine lambda[i] which are roots of the  characteristic equations?in other word how i can build characteristic relation for coupling equation?

2)how i can gain value for alpha[i] and GAMMA[i]

by using equation

Q1 := subs(x = 0, sum(H[i]*exp(lambda[i]*x), i = 1 .. 8)); Q2 := subs(x = L, sum(H[i]*exp(lambda[i]*x), i = 1 .. 8)); Q3 := subs(x = 0, sum(alpha[i]*H[i]*exp(lambda[i]*x), i = 1 .. 8)); Q4 := subs(x = L, sum(alpha[i]*H[i]*exp(lambda[i]*x), i = 1 .. 8)); Q5 := subs(x = 0, sum(GAMMA[i]*H[i]*exp(lambda[i]*x), i = 1 .. 8)); Q6 := subs(x = L, sum(GAMMA[i]*H[i]*exp(lambda[i]*x), i = 1 .. 8)); M := diff(sum(GAMMA[i]*H[i]*exp(lambda[i]*x), i = 1 .. 8), x); Q7 := subs(x = 0, M); Q8 := subs(x = L, M)

????

thanks...

chebyshev.mw