By Erwin Kreyszig
The 10th variation of this bestselling textual content comprises examples in additional element and extra utilized workouts; either adjustments are geared toward making the cloth extra appropriate and available to readers. Kreyszig introduces engineers and laptop scientists to complicated math subject matters as they relate to functional difficulties. It is going into the next themes at nice depth differential equations, partial differential equations, Fourier research, vector research, complicated research, and linear algebra/differential equations.
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Additional info for Advanced Engineering Mathematics, 10th Edition
Modeling EXAMPLE 2 13 Separable ODE The ODE y r ϭ (x ϩ 1)e؊xy 2 is separable; we obtain y ؊2 dy ϭ (x ϩ 1)e؊x dx. By integration, EXAMPLE 3 Ϫy ؊1 ϭ Ϫ(x ϩ 2)e؊x ϩ c, yϭ 1 . (x ϩ 2)eϪx Ϫ c Initial Value Problem (IVP). 8. Solution. By separation and integration, dy y ϭ Ϫ2x dx, ln y ϭ Ϫx 2 ϩ ෂ c, y ϭ ce؊x . 2 This is the general solution. 8. 8e؊x . This is a particular solution, representing a bell-shaped curve (Fig. 10). y 1 –2 –1 0 1 2 x Fig. 10. Solution in Example 3 (bell-shaped curve) Modeling The importance of modeling was emphasized in Sec.
Suppose that the time rate of change is the difference between a sinusoidal input of a 24-hour period from the thyroid gland and a continuous removal rate proportional to the level present. Set up a model for the hormone level in the blood and find its general solution. Find the particular solution satisfying a suitable initial condition. Solution. Step 1. Setting up a model. Let y(t) be the hormone level at time t. Then the removal rate is Ky(t). The input rate is A ϩ B cos vt, where v ϭ 2p>24 ϭ p>12 and A is the average input rate; here A м B to make the input rate nonnegative.
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. 3) M(x, y) dx ϩ N(x, y) dy ϭ 0 (1) is called an exact differential equation if the differential form M(x, y) dx ϩ N(x, y) dy is exact, that is, this form is the differential du ϭ (2) 0u 0u dx ϩ dy 0x 0y of some function u(x, y).
Advanced Engineering Mathematics, 10th Edition by Erwin Kreyszig