By Erwin Kreyszig
The 10th variation of this bestselling textual content contains examples in additional element and extra utilized workouts; either adjustments are geared toward making the cloth extra suitable and available to readers. Kreyszig introduces engineers and desktop scientists to complex math issues as they relate to functional difficulties. It is going into the subsequent themes at nice depth differential equations, partial differential equations, Fourier research, vector research, advanced research, and linear algebra/differential equations.
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Additional info for Advanced Engineering Mathematics (10th Edition)
01(y Ϫ 5000). (4) Step 2. Solution of the model. The ODE (4) is separable. 01t. 01t ϩ c*, Initially the tank contains 100 lb of salt. Hence y(0) ϭ 100 is the initial condition that will give the unique solution. Substituting y ϭ 100 and t ϭ 0 in the last equation gives 100 Ϫ 5000 ϭ ce0 ϭ c. Hence c ϭ Ϫ4900. 01t. (5) This function shows an exponential approach to the limit 5000 lb; see Fig. 11. Can you explain physically that y(t) should increase with time? That its limit is 5000 lb? Can you see the limit directly from the ODE?
Particular solution in Example 3 Reduction to Linear Form. Bernoulli Equation Numerous applications can be modeled by ODEs that are nonlinear but can be transformed to linear ODEs. One of the most useful ones of these is the Bernoulli equation7 (9) y r ϩ p(x)y ϭ g(x)y a (a any real number). 7 JAKOB BERNOULLI (1654–1705), Swiss mathematician, professor at Basel, also known for his contribution to elasticity theory and mathematical probability. The method for solving Bernoulli’s equation was discovered by Leibniz in 1696.
A first-order ODE is said to be linear if it can be brought into the form (1) y r ϩ p(x)y ϭ r(x), by algebra, and nonlinear if it cannot be brought into this form. The defining feature of the linear ODE (1) is that it is linear in both the unknown function y and its derivative y r ϭ dy>dx, whereas p and r may be any given functions of x. If in an application the independent variable is time, we write t instead of x. If the first term is f (x)y r (instead of y r ), divide the equation by f (x) to get the standard form (1), with y r as the first term, which is practical.