Lecture Notes in Differential Equations - Bruce E. Shapiro
Lecture Notes in Differential Equations - Bruce E. Shapiro Lecture Notes in Differential Equations - Bruce E. Shapiro
416 APPENDIX C. SUMMARY OF METHODS To solve a(t)y ′′ + b(t)y ′ + c(t)y = g(t) about a point t 0 where a(t) = 0 but the limits b(t)/a(t) and c(t)/a(t) exist as t → 0 (a regular singularity), solve the indicial equation r(r − 1) + rp 0 + q 0 = 0 for r where p 0 = lim t→0 b(t 0 )/a(t 0 ) and and q 0 = lim t→0 c(t 0 )/a(t 0 ). Then one solution to the homogeneous equation is y(t) = (t − t 0 ) r ∞ ∑ k=0 c k (t − t 0 ) k for some unknown coefficients c k .Determine the coefficients by linear independance of the powers of t. The second solution is found by reduction of order and the particular solution by variation of parameters. Method of Frobenius To solve (t − t 0 ) 2 y ′′ + (t − t 0 )p(t)y ′ + q(t)y = 0 where p and q are analytic at t 0 , let p 0 = p(0) and q 0 = q(0) and find the roots α 1 ≥ α 2 of α 2 + (p 0 − 1)α + q 0 = 0 Define ∆ = α 1 − α 2 . Then for some unknowns c k , a first solution is y 1 (t) = (t − t 0 ) α1 ∞ ∑ k=0 c k (t − t 0 ) k If ∆ ∈ R is not an integer or the roots are complex, y 2 (t) = (t − t 0 ) α2 ∞ ∑ k=0 a k (t − t 0 ) k If α 1 = α 2 = α, then y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α If ∆ ∈ Z, then y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α2 ∞ ∑ k=0 ∞ ∑ k=0 a k (t − t 0 ) k a k (t − t 0 ) k
Bibliography [1] Bear, H.S. Differential Equations. Addison-Wesley. (1962) Has a very good description of Picard iteration that many students find helpful. Unfortunately this text is long since out of print so you’ll have to track it down in the library. [2] Boyce, William E and DiPrima, Richard C. Elementary Differential Equations. 9th Edition. Wiley. (2008) This is the classic introductory textbook on differential equations. It also has a longer version with two additional chapters that cover boundary value problems but is otherwise identical. It is at the same level as [12] and [9] and the choice of which book you read really depends on your style. The first edition of this book was the best; the later editions have only served to fill the coffers of the publishers, while the later editions use heavy paper and are quite expensive. Each edition just gets heavier and heavier. [3] Bronson, Richard and Costa Gabriel. Schaum’s Outline of Differential Equations. 4th Edition. McGraw-Hill. (2006) The usual from Schaum’s. A good, concise, low cost outline of the subject. [4] Churchill, Ruel V. Operational Mathematics. 3rd Edition McGraw-Hill. (1972) Classic textbook on Laplace and Fourier methods. [5] Coddington, Earl A. An Introduction to Ordinary Differential Equations. Dover Books (1989). This is a reprint of an earlier edition. It contains the classical theory of differential equations with all of the proofs at a level accessible to advanced undergraduates with a good background in calculus. (Analysis is not a prerequisite to this textbook). Lots of bang for your buck. [6] Coddington, Earl A. Levinson, Norman. Theory of Ordinary Differential Equations. Krieger (1984) (Reprint of earlier edition). The standard graduate text in classical differential equations written in the 417
- Page 373 and 374: 365 Figure 33.3: Illustration of th
- Page 375 and 376: 367 result with a smaller step size
- Page 377 and 378: 369 Expanding the final term in a T
- Page 379 and 380: 371 k 2 = y 0 + h 2 f(t 0, k 1 ) (3
- Page 381 and 382: Lesson 34 Critical Points of Autono
- Page 383 and 384: 375 Since both f and g are differen
- Page 385 and 386: 377 Using the cos π/4 = √ 2/2 an
- Page 387 and 388: 379 values, of the matrix. We find
- Page 389 and 390: 381 Distinct Real Nonzero Eigenvalu
- Page 391 and 392: 383 eigendirection {λ 1 , v 1 }dom
- Page 393 and 394: 385 Figure 34.5: Phase portraits ty
- Page 395 and 396: 387 Complex Conjugate Pair with non
- Page 397 and 398: 389 The angular change is described
- Page 399 and 400: 391 Figure 34.8: Topological instab
- Page 401 and 402: 393 Figure 34.10: phase portraits f
- Page 403 and 404: Appendix A Table of Integrals Basic
- Page 405 and 406: 397 ∫ x √ x − adx = 2 3 a(x
- Page 407 and 408: 399 ∫ x √ ax2 + bx + c dx = 1 a
- Page 409 and 410: 401 ∫ ∫ ∫ ∫ e ax2 dx = −
- Page 411 and 412: 403 ∫ tan 3 axdx = 1 a ln cos ax
- Page 413 and 414: 405 Products of Trigonometric Funct
- Page 415 and 416: Appendix B Table of Laplace Transfo
- Page 417 and 418: 409 e at cosh kt t sin kt t cos kt
- Page 419 and 420: Appendix C Summary of Methods First
- Page 421 and 422: 413 The resulting equation is linea
- Page 423: 415 for y once z is known. Method o
- Page 427 and 428: BIBLIOGRAPHY 419
- Page 429 and 430: BIBLIOGRAPHY 421
Bibliography<br />
[1] Bear, H.S. <strong>Differential</strong> <strong>Equations</strong>. Addison-Wesley. (1962) Has a very<br />
good description of Picard iteration that many students f<strong>in</strong>d helpful.<br />
Unfortunately this text is long s<strong>in</strong>ce out of pr<strong>in</strong>t so you’ll have to track<br />
it down <strong>in</strong> the library.<br />
[2] Boyce, William E and DiPrima, Richard C. Elementary <strong>Differential</strong><br />
<strong>Equations</strong>. 9th Edition. Wiley. (2008) This is the classic <strong>in</strong>troductory<br />
textbook on differential equations. It also has a longer version with<br />
two additional chapters that cover boundary value problems but is<br />
otherwise identical. It is at the same level as [12] and [9] and the choice<br />
of which book you read really depends on your style. The first edition<br />
of this book was the best; the later editions have only served to fill the<br />
coffers of the publishers, while the later editions use heavy paper and<br />
are quite expensive. Each edition just gets heavier and heavier.<br />
[3] Bronson, Richard and Costa Gabriel. Schaum’s Outl<strong>in</strong>e of <strong>Differential</strong><br />
<strong>Equations</strong>. 4th Edition. McGraw-Hill. (2006) The usual from Schaum’s.<br />
A good, concise, low cost outl<strong>in</strong>e of the subject.<br />
[4] Churchill, Ruel V. Operational Mathematics. 3rd Edition McGraw-Hill.<br />
(1972) Classic textbook on Laplace and Fourier methods.<br />
[5] Codd<strong>in</strong>gton, Earl A. An Introduction to Ord<strong>in</strong>ary <strong>Differential</strong> <strong>Equations</strong>.<br />
Dover Books (1989). This is a repr<strong>in</strong>t of an earlier edition. It<br />
conta<strong>in</strong>s the classical theory of differential equations with all of the<br />
proofs at a level accessible to advanced undergraduates with a good<br />
background <strong>in</strong> calculus. (Analysis is not a prerequisite to this textbook).<br />
Lots of bang for your buck.<br />
[6] Codd<strong>in</strong>gton, Earl A. Lev<strong>in</strong>son, Norman. Theory of Ord<strong>in</strong>ary <strong>Differential</strong><br />
<strong>Equations</strong>. Krieger (1984) (Repr<strong>in</strong>t of earlier edition). The standard<br />
graduate text <strong>in</strong> classical differential equations written <strong>in</strong> the<br />
417
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