Linear system of ordinary differential equations with constant coefficients example

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