Theory of Systems of Linear Differential Equations It turns out that the theory of systems of linear differential equations resembles the theory of higher order differential equations. This discussion will adopt the following notation. Consider the system of differential equations x_{1}'
= p_{11}(t) x_{1} + ... + p_{1n}(t) x_{n}
+ g_{1}(t) We write this system as x' = P(t)x + g(t) A vector x = f(t) is a solution of the system of differential equation if f' = P(t)f + g(t) If g(t) = 0 the system of differential equations is called homogeneous. Otherwise, it is called nonhomogeneous.
Just as we had the Wronskian for higher order linear differential equations, we can define a similar beast for systems of linear differential equations. If x^{(1)}, x^{(2)}, ..., x^{(n)} are n solutions of an n x n system, then the Wronskian of this set is the determinant of the matrix whose i^{th} column is x^{(i)}.
Example Let
Then
It is a direct consequence from linear algebra that solutions are linearly independent if and only if the Wronskian is nonzero. In fact, more is true. There is a generalizations of Abel's theorem for systems of linear differential equations.
dW
The main theorem on uniqueness and existence of solutions of systems of differential equations also holds true. We state it below.
In particular, if x^{(1)}, x^{(2)}, ..., x^{(n)} are solutions of the homogeneous system, and if the Wronskian is nonzero, then y = c_{1}x^{(1)} + c_{2}x^{(2)} + ... + c_{k}x^{(k)} is the general solution to the system. We call x^{(1)}, x^{(2)}, ..., x^{(n)} a fundamental set of solutions to the system of differential equations. In particular, if the Wronskian matrix at t_{0} is the identity matrix (W(t_{0}) = I) then its determinant is one hence not zero. This gives us the following theorem
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