The general solution of an inhomogeneous system is the sum of the general solution of a homogeneous system and some particular solution of an inhomogeneous system.
To find a general solution to an inhomogeneous system, you can apply the Lagrange method of variation of arbitrary constants.
Let us consider a linear homogeneous system of ordinary differential equations of the form
which in vector form is written as
Matrix Φ , the columns of which are n linearly independent solutions Y1(x), Y2(x), ..., Yn(x) of a homogeneous linear system Y" = A(x)Y is called the fundamental matrix of solutions of the system:
The fundamental matrix of solutions of a homogeneous linear system Y" = A(x)Y satisfies the matrix equation Φ" = A(x)Φ.
Recall that the Wronski determinant of linearly independent solutions Y1(x), Y2(x), ..., Yn(x) is nonzero by .
Consider a linear system of nth order differential equations:
A linear system is Lyapunov stable for t ≥ t0 if each of its solutions x = φ(t) is Lyapunov stable for t ≥ t0.
A linear system is asymptotically Lyapunov stable as t → ∞ if each of its solutions x = φ(t) is Lyapunov stable as t → ∞.
The solutions of a linear system are either all stable at the same time or all unstable. The following statements are true.
Theorem on the stability of solutions to a linear system of differential equations. Let in the inhomogeneous linear system x" = A(t)x + b(t) the matrix A(t) and the vector function b(t) be continuous on the interval )
