By Erwin Kreyszig
Advanced Engineering arithmetic, tenth Edition is understood for its entire assurance, cautious and proper arithmetic, remarkable workouts, and self-contained material components for optimum flexibility. the hot variation maintains with the culture of delivering teachers and scholars with a accomplished and updated source for educating and studying engineering arithmetic, that's, utilized arithmetic for engineers and physicists, mathematicians and machine scientists, in addition to participants of different disciplines.
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Extra resources for Advanced engineering mathematics
Thus With this F and h r ϭ p, Eq. (1*) becomes ehy r ϩ h r ehy ϭ ehy r ϩ (eh) r y ϭ (ehy) r ϭ reh. By integration, ehy ϭ Ύ e r dx ϩ c. h Dividing by eh, we obtain the desired solution formula (4) Ύ y(x) ϭ e؊h a ehr dx ϩ cb, hϭ Ύ p(x) dx. SEC. 5 Linear ODEs. Bernoulli Equation. Population Dynamics 29 This reduces solving (1) to the generally simpler task of evaluating integrals. For ODEs for which this is still difficult, you may have to use a numeric method for integrals from Sec. 5 or for the ODE itself from Sec.
5x , y(0) ϭ p 11. y r ϭ y ϩ ex, y ϭ (x ϩ c)ex, y(0) ϭ 12 12. yy r ϭ 4x, y 2 Ϫ 4x 2 ϭ c (y Ͼ 0), y(1) ϭ 4 1 13. 25 1 ϩ ce؊x 14. y r tan x ϭ 2y Ϫ 8, y ϭ c sin 2 x ϩ 4, y(12 p) ϭ 0 15. Find two constant solutions of the ODE in Prob. 13 by inspection. 16. Singular solution. An ODE may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. The ODE y r 2 Ϫ xy r ϩ y ϭ 0 is of this kind. Show by differentiation and substitution that it has the general solution y ϭ cx Ϫ c2 and the singular solution y ϭ x 2>4.
Physical information. The rate of change y r (x) is proportional to the pressure. At 18,000 ft it is half its value y0 ϭ y(0) at sea level. Hint. Remember from calculus that if y ϭ ekx, then y r ϭ kekx ϭ ky. Can you see without calculation that the answer should be close to y0>4? SEC. 2 Geometric Meaning of yЈ ϭ ƒ(x, y). 2 9 Geometric Meaning of y r ϭ f (x, y). Direction Fields, Euler’s Method A first-order ODE y r ϭ f (x, y) (1) has a simple geometric interpretation. From calculus you know that the derivative y r (x) of y(x) is the slope of y(x).