How to find roots of a quartic function with unknown constant coefficients?

The polynomial is:

x^4 + 10.1ax^3 + 1.81x^2 + 9ax + 0.81 = 0

where a is an unkown constant coefficient.

I have to find the corresponding x values to further solve an ODE.

Answer 1

[James Tursa]

You can look here:

https://en.wikipedia.org/wiki/Quartic_function

But the nature of the roots is going to depend on the value of a. Do you know anything at all about a? What is the context of the overall problem you are solving? What is the ODE?

[Hrinay]

No it just represents a dampening coefficient for a spring mass system. The ode models the movement of a building. Essentially a fbd with 2 masses, 2 springs, and 2 dampeners. I get two odes that model the forces on the mass block. I am trying to solve the simultaneous set of ODE's using cramers rule than MUC because the odes all have constant coefficients of which we have a value for every constant except the dampening coefficients C1 and C2.

The FBD:

Answer 2

[Torsten]

syms x a
p = x^4 + 10.1*a*x^3 + 1.81*x^2 + 9*a*x + 0.81;
sol = solve(p==0,'MaxDegree',4)

sol = 

vpa(subs(sol,a,1))   % E.g. for a = 1

ans = 

[Walter Roberson]

The roots get complicated and fragile

filename = fullfile(tempdir, 'sol.m');
addpath(tempdir);
syms x a
p = x^4 + 10.1*a*x^3 + 1.81*x^2 + 9*a*x + 0.81;
sol = solve(p==0);
solF = matlabFunction(sol, 'file', filename)

solF = function_handle with value:
    @sol

A = linspace(-1/2,1/2,200);
Y = cell2mat(arrayfun(solF, A, 'uniform', 0));
figure();
plot(A, real(Y(1,:)), A, real(Y(2,:)), A, real(Y(3,:)), A, real(Y(4,:)));
legend({'1 real', '2 real', '3 real', '4 real'})

figure();
plot(A, imag(Y(1,:)), A, imag(Y(2,:)), A, imag(Y(3,:)), A, imag(Y(4,:)));
legend({'1 imag', '2 imag', '3 imag', '4 imag'})