By Paul Waltman, Mathematics

ISBN-10: 0127339108

ISBN-13: 9780127339108

**Read Online or Download A Second Course in Elementary Differential Equations PDF**

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

**Example text**

A) = ρ(λ). If A£ is a complex eigenvalue, then Pi^i) = 0, which implies thatp^) = 0 or that Xf is an eigenvalue. Thus, complex eigenvalues occur as complex-conjugate pairs. Now let Ci be an eigenvector corresponding to the complex eigenvalue kt. Then, since {A — X^Ci = 0, (A - V ) c ! = 0 or (A - X,/)^ = 0. Since λι is an eigenvalue of A, c£ is an eigenvector. Thus, knowing a complex eigenvalue λ and its corresponding eigenvector c lets us determine a second eigenvalue-eigenvector pair, X, c. In effect, taking the real and imaginary parts of a complex solution eA

This process can be continued until r1(t), . . , rn{t) are found merely by solvingfirst-orderlinear differential equations. Proof. Let Φ(ή = Σΐ=ο0+ι(0^· The idea of the proof is to show that Φ(ή satisfies Φ' = ΑΦ, Φ(0) = / so that Φ(/) = eA\ by the uniqueness of so lutions. For convenience, define r0(t) = 0. Then j=o = "f[Vi O+i (0 + 0(0]/} j=o so that Φ'(0 - Λ„Φ = "£ ( V i 0>i (0 + rj(t))Pj - j / f rj+1(t)Pj j=0 j=0 = Σ ( V I - K)r^{t)Pj + "f 0(0^5 = ? ( V i - K)rj+i(t)Pj + "l O+i(0^1. 2), the last line may be rewritten as Φ'(/) - kMt) = Σ 2 [( Α - Vi 7 )/} + (Vi - Wlo+iW J=0 = °Σ (A - knI)Pjrj+1(t) j=o = (^-ν)Σ2/;·ο+ι(ο.

9. Find a fundamental matrix for x' = Ax for each A given in Exercise 7. 10. (a) Derive the Taylor series expansion for f(6) = ew. ) (b) Rearrange the series in (a) into real and purely imaginary parts (each part will be a series). (c) Identify the series in (b) and deduce Euler's formula, e>ie = cos(0) + /sin(0). (d) What do you need to know about the convergence of a series to perform the rearrangement in (b)? 7. THE CONSTANT COEFFICIENT CASE: THE PUTZER ALGORITHM 49 7. The Constant Coefficient Case: The Putzer Algorithm The analysis of the preceding section depended on finding either n distinct eigenvalues or sufficient linearly independent eigenvectors when an eigenvalue corresponded to a repeated root of the characteristic polynomial.

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

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