# A Second Course in Elementary Differential Equations by Paul Waltman, Mathematics

By Paul Waltman, Mathematics

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**Additional info for A Second Course in Elementary Differential Equations**

**Sample text**

Rather than attempt to find the Γ, we reason as follows: eBt has entries eki\ Premultiphcation (multiplication on the left) by Γ"1 and postmultiplication (multiplication on the right) by T rearranges and combines these. 1) of the form [cYekit c2eXit y = _cne^_ [~ci] = eXitc, = eXtt _cn\ 5. THE CONSTANT COEFFICIENT CASE: REAL AND DISTINCT EIGENVALUES 31 where Af is one of the diagonal elements of B = TAT l. Then ~kiCie^~ = Ky, lfne kit or hy = Ay, which we write as (A - XJ)y = 0. 5). The (complex) numbers λ such that detG4-A/) = 0, are called the eigenvalues of the matrix A.

Given a sequence of matrices An, we can form another sequence (called the sequence of partial sums) by defining Sn = Ax + · · · + An. We denote the se quence {Sn} by £ " Ä 1 Ai and call J ^ Ai an infinite series. If limM_00 Sn = S, then the series is said to converge and its sum is defined to be S. If {Sn} does not converge, the series is said to diverge, and the sum is not defined. It is important to note that we can often show that a series converges without being able to find the limit. For example, we could investigate the (real) infinite series 00 νΠ o ni or the sequence of partial sums x2 xn and deduce that it converges.

Since all initial conditions can be satisfied, given a fundamental matrix 0,real solutions are of the form 0 ( t ) c , where 0 ( t )and c may have complex entries. Representing a real vector as the product of a matrix with complex entries and a constant vector with complex entries is, at least, inelegant and frequently may be awkward. For this reason we seek a way to find a real fundamental matrix. That this can always be done is a consequence of the following theorem. 6. 1). Proof. The complex-valued function φ(ί) can be written as q>(t) = u(t) + iv(t) where u(i) and v(t) are real-valued functions (u(t) = Re φ(ή, v(t) = lm