Lecture Notes in Differential Equations - Bruce E. Shapiro
Lecture Notes in Differential Equations - Bruce E. Shapiro Lecture Notes in Differential Equations - Bruce E. Shapiro
400 APPENDIX A. TABLE OF INTEGRALS ∫ x ln ( a 2 − b 2 x 2) dx = − 1 2 x2 + 1 2 (x 2 − a2 b 2 ) ln ( a 2 − b 2 x 2) (A.49) Integrals with Exponentials ∫ e ax dx = 1 a eax (A.50) ∫ √xe ax dx = 1 √ xe ax + i√ π a 2a erf ( i √ ax ) , 3/2 where erf(x) = √ 2 ∫ x e −t2 dt π ∫ xe x dx = (x − 1)e x 0 (A.51) (A.52) ∫ ∫ xe ax dx = ( x a − 1 a 2 ) e ax x 2 e x dx = ( x 2 − 2x + 2 ) e x (A.53) (A.54) ∫ ∫ x 2 e ax dx = ( x 2 a − 2x a 2 + 2 a 3 ) e ax x 3 e x dx = ( x 3 − 3x 2 + 6x − 6 ) e x (A.55) (A.56) ∫ x n e ax dx = xn e ax − n ∫ a a x n−1 e ax dx (A.57) ∫ x n e ax dx = (−1)n Γ[1 + n, −ax], an+1 where Γ(a, x) = ∫ ∞ x t a−1 e −t dt (A.58)
401 ∫ ∫ ∫ ∫ e ax2 dx = − i√ π 2 √ a erf ( ix √ a ) (A.59) e −ax2 dx = √ π 2 √ a erf ( x √ a ) (A.60) xe −ax2 dx = − 1 2a e−ax2 (A.61) x 2 e −ax2 dx = 1 4√ π a 3 erf(x√ a) − x 2a e−ax2 (A.62) Integrals with Trigonometric Functions ∫ sin axdx = − 1 cos ax a (A.63) ∫ sin 2 axdx = x 2 − sin 2ax 4a (A.64) ∫ sin n axdx = − 1 [ 1 a cos ax 2F 1 2 , 1 − n , 3 ] 2 2 , cos2 ax (A.65) ∫ sin 3 3 cos ax cos 3ax axdx = − + 4a 12a ∫ cos axdx = 1 sin ax a (A.67) (A.66) ∫ cos 2 axdx = x 2 + sin 2ax 4a (A.68) ∫ [ cos p 1 1 + p axdx = − a(1 + p) cos1+p ax × 2 F 1 2 , 1 2 , 3 + p ] 2 , cos2 ax (A.69) ∫ cos 3 axdx = 3 sin ax 4a + sin 3ax 12a (A.70)
- Page 357 and 358: 349 Summary of Translation Formulas
- Page 359 and 360: 351 The inverse transform is [ ] f(
- Page 361 and 362: 353 Example 32.18. Find the Laplace
- Page 363 and 364: 355 Similarly, we can express a uni
- Page 365 and 366: 357 Figure 32.7: Solution of exampl
- Page 367 and 368: Lesson 33 Numerical Methods Euler
- Page 369 and 370: 361 Figure 33.1: Illustration of Eu
- Page 371 and 372: 363 y 4 = y 3 + hf(t 3 , y 3 ) (33.
- 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: 399 ∫ x √ ax2 + bx + c dx = 1 a
- 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 and 424: 415 for y once z is known. Method o
- Page 425 and 426: Bibliography [1] Bear, H.S. Differe
- Page 427 and 428: BIBLIOGRAPHY 419
- Page 429 and 430: BIBLIOGRAPHY 421
400 APPENDIX A. TABLE OF INTEGRALS<br />
∫<br />
x ln ( a 2 − b 2 x 2) dx = − 1 2 x2 + 1 2<br />
(x 2 − a2<br />
b 2 )<br />
ln ( a 2 − b 2 x 2)<br />
(A.49)<br />
Integrals with Exponentials<br />
∫<br />
e ax dx = 1 a eax<br />
(A.50)<br />
∫ √xe ax dx = 1 √ xe ax + i√ π<br />
a 2a erf ( i √ ax ) ,<br />
3/2<br />
where erf(x) = √ 2 ∫ x<br />
e −t2 dt<br />
π<br />
∫<br />
xe x dx = (x − 1)e x<br />
0<br />
(A.51)<br />
(A.52)<br />
∫<br />
∫<br />
xe ax dx =<br />
( x<br />
a − 1 a 2 )<br />
e ax<br />
x 2 e x dx = ( x 2 − 2x + 2 ) e x<br />
(A.53)<br />
(A.54)<br />
∫<br />
∫<br />
x 2 e ax dx =<br />
( x<br />
2<br />
a − 2x<br />
a 2 + 2 a 3 )<br />
e ax<br />
x 3 e x dx = ( x 3 − 3x 2 + 6x − 6 ) e x<br />
(A.55)<br />
(A.56)<br />
∫<br />
x n e ax dx = xn e ax<br />
− n ∫<br />
a a<br />
x n−1 e ax dx<br />
(A.57)<br />
∫<br />
x n e ax dx = (−1)n Γ[1 + n, −ax],<br />
an+1 where Γ(a, x) =<br />
∫ ∞<br />
x<br />
t a−1 e −t dt<br />
(A.58)
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