SOLVE LINEAR DIFERENTIAL EQUATION WITH ONE UNKNOWN

I have the system of the image, but i want to solve the exatly system but insted of a fixed number where the four goes, i would like to find the range of solutions but with the unknown where the four goes.

I would usually solve this with pencil and paper but my system is 6x6, Thanks in Advance.

AUTHOR edit:

Sorry i think my explanation was not clear, what i would like to solve this ->

eqn1 = du == 3*u + X*v;

eqn2 = dv == -X*u + 3*v;

where X is a parameter, and i would like to find the family of solution's in function of my parameter X.

Answer 1

[Walter Roberson]

syms u(t) v(t)
du = diff(u,t);
dv = diff(v,t);
eqn1 = du == 3*u + 4*v;
eqn2 = dv == -4*u + 3*v;
sol = dsolve([eqn1, eqn2])

sol = struct with fields:
    v: [1×1 sym]
    u: [1×1 sym]

sol.u

ans = 

sol.v

ans = 

[Tomás Yamamoto]

Sorry i think my explanation was not clear, what i would like to solve this ->

eqn1 = du == 3*u + X*v;

eqn2 = dv == -X*u + 3*v;

where X is a parameter, and i would like to find the family of solution's in function of my parameter X.

[Walter Roberson]

syms u(t) v(t) X
du = diff(u,t);
dv = diff(v,t);
eqn1 = du == 3*u + X*v;
eqn2 = dv == -X*u + 3*v;
sol = dsolve([eqn1, eqn2])

sol = struct with fields:
    v: [1×1 sym]
    u: [1×1 sym]

simplify(sol.u)

ans = 

simplify(sol.v)

ans =