By Erwin Kreyszig
The 10th variation of this bestselling textual content comprises examples in additional element and extra utilized routines; either alterations are geared toward making the fabric extra correct and available to readers. Kreyszig introduces engineers and machine scientists to complicated math themes as they relate to functional difficulties. It is going into the next subject matters at nice depth differential equations, partial differential equations, Fourier research, vector research, complicated research, and linear algebra/differential equations.
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Extra info for Advanced Engineering Mathematics (10th Edition)
Qxd 7/30/10 8:15 PM Page 9 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). Hence a solution curve of (1) that passes through a point (x 0, y0) must have, at that point, the slope y r (x 0) equal to the value of f at that point; that is, y r (x 0) ϭ f (x 0, y0). Using this fact, we can develop graphic or numeric methods for obtaining approximate solutions of ODEs (1).
16. 15. The result should surprise you! S Small portion of rope Δ S + ΔS Fig. 16. Problem 33 34. TEAM PROJECT. Family of Curves. A family of curves can often be characterized as the general solution of y r ϭ f (x, y). (a) Show that for the circles with center at the origin we get y r ϭ Ϫx>y. (b) Graph some of the hyperbolas xy ϭ c. Find an ODE for them. (c) Find an ODE for the straight lines through the origin. (d) You will see that the product of the right sides of the ODEs in (a) and (c) equals Ϫ1.
Exact ODEs. Integrating Factors We recall from calculus that if a function u(x, y) has continuous partial derivatives, its differential (also called its total differential) is du ϭ 0u 0u dx ϩ dy. 0x 0y From this it follows that if u(x, y) ϭ c ϭ const, then du ϭ 0. qxd 7/30/10 8:15 PM Page 21 SEC. 4 Exact ODEs. Integrating Factors 21 an ODE that we can solve by going backward. This idea leads to a powerful solution method as follows. A first-order ODE M(x, y) ϩ N(x, y)y r ϭ 0, written as (use dy ϭ y r dx as in Sec.