Formal Solutions

The principal objects of study in this book are linear homogeneous ordinary differential equations that involve one scalar parameter. By introducing the higher derivatives of the dependent variable or variables as additional unknown functions, any such equation, or system of equations, can be rewritten as an equivalent first-order system: 2.1-1$$\begin{array}{*{20}{c}} {\frac{{d{y_j}}}{{dx}} = \sum\limits_{k = 1}^n {{a_{jk}}(x, \in } ){y_k},}&{j = 1,2, \ldots ,n.} \end{array}$$ The parameter has been designated by the letter ∈, because it will be small in most situations to be encountered. If the n X n matrix A (x, ∈) with entries a _jk (x, ∈) and the column vector y with components y _j , j = 1, 2,...,n, are introduced, equation (2.1-1) takes on the concise and convenient form 2.1-2$$y\prime = A(x, \in )y.$$ The prime indicates differentiation with respect to x.