Variation of Parameters

Brown University, Applied Mathematics

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Variation of Parameters

 

Suppose that we know a fundamental matrix \( {\bf X} (t) \) of the vector system of homogeneous linear differential equations:

\[ \dot{\bf x} = {\bf P} (t)\, {\bf x} + {\bf f}(t) , \]
Here \( {\bf P} (t) \) is \( n\times n \) matrix with continuous entries, \( {\bf x} (t) \) is an \( n- \) column vector of unknown functions to be determined, and \( {\bf f} (t) \) is given driven column vector.

The variation of parameters method suggests to represent a particular solution of the given nonhomogeneous system of differential equations in the form
\[ {\bf x}_p (t) = {\bf X} (t)\, {\bf u}(t) . \]
To determine the unknown column-vector \( {\bf u} (t), \) we substitute this form of solution into the driven vector equation to obtain
\[ \dot{\bf x}_p (t) = \dot{\bf X} (t)\,{\bf u}(t) + {\bf X} (t)\, \dot{\bf u}(t) = {\bf P} (t)\,{\bf X} (t)\, {\bf u}(t) + {\bf f} (t). \]
Since \( \dot{\bf X} (t) = {\bf P} (t)\, {\bf X}(t), \) we have
\[ {\bf X} (t)\, \dot{\bf u}(t) = {\bf f} (t) \qquad\mbox{or} \qquad \dot{\bf u}(t) = {\bf X}^{-1} (t)\, {\bf f} (t) . \]
Integration yields
\[ {\bf u} (t) = \int {\bf X}^{-1} (t)\,{\bf f}(t) \, {\text d}t + {\bf c} , \]
where \( {\bf c} \) is a column vector of arbitrary constants of integration. Then the general solution of the given driven system of differential equations becomes
\[ {\bf x} (t) = {\bf X} (t)\,\int {\bf X}^{-1} (t)\,{\bf f}(t) \, {\text d}t + {\bf X} (t)\,{\bf c} . \]
Here the term \( {\bf x}_h (t) = {\bf X} (t)\,{\bf c} \) is the general solution of the corresponding homogeneous vector equation, \( \dot{\bf x} (t) = {\bf P} (t)\,{\bf x}, \) and \( {\bf x}_p (t) = {\bf X} (t)\,\int {\bf X}^{-1} (t)\,{\bf f}(t) \, {|text d}t \) is a particular solution of the nonhomogeneous vector equation \( \dot{\bf x} (t) = {\bf P} (t)\,{\bf x} + {\bf f} (t). \)

 

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