We will Resolve, in this section a linear ODE's system whose roots of the auxiliary equation are complex
Lets the ODE's system
con
Note that this is actually an initial value problem represented by a linear system of differential equations
where, x is the independent variable and the variables y, z are dependent variables
To begin, and as explained in the theory sections, we seek solutions of the form
We substitute (1) in our system to obtain
This system will have nontrivial solution if and only if determinant of their coefficients is annulled, ie
This gives a quadratic equation in the m- variable:
This is the case of complex conjugate roots, we apply methods saw in theory sections, using the initial conditions and taking constants 1, we obtain
Substituting in the system we have finally the general solution of our ODEs system