By Erwin Kreyszig
The 10th variation of this bestselling textual content contains examples in additional aspect and extra utilized routines; either alterations are aimed toward making the cloth extra appropriate and obtainable to readers. Kreyszig introduces engineers and desktop scientists to complicated math subject matters 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)
Integrating Factors 23 Step 3. Checking an implicit solution. We can check by differentiating the implicit solution u(x, y) ϭ c implicitly and see whether this leads to the given ODE (7): (9) du ϭ 0u 0u dx ϩ dy ϭ cos (x ϩ y) dx ϩ (cos (x ϩ y) ϩ 3y 2 ϩ 2y) dy ϭ 0. 0x 0y This completes the check. EXAMPLE 2 An Initial Value Problem Solve the initial value problem (cos y sinh x ϩ 1) dx Ϫ sin y cosh x dy ϭ 0, (10) Solution. y(1) ϭ 2. You may verify that the given ODE is exact. We find u. For a change, let us use (6*), Ύ u ϭ Ϫ sin y cosh x dy ϩ l(x) ϭ cos y cosh x ϩ l(x).
Then by Newton’s law, dT ϭ k(T Ϫ TA). dt (6) Such experimental laws are derived under idealized assumptions that rarely hold exactly. However, even if a model seems to fit the reality only poorly (as in the present case), it may still give valuable qualitative information. To see how good a model is, the engineer will collect experimental data and compare them with calculations from the model. Step 2. General solution. We cannot solve (6) because we do not know TA, just that it varied between 50°F and 40°F, so we follow the Golden Rule: If you cannot solve your problem, try to solve a simpler one.
Hence in this case the exact equation (13) is FP dx ϩ FQ dy ϭ Ϫy dx ϩ x dy x 2 y ϭ d a b ϭ 0. x Solution y x ϭ c. These are straight lines y ϭ cx through the origin. ) It is remarkable that we can readily find other integrating factors for the equation Ϫy dx ϩ x dy ϭ 0, namely, 1>y 2, 1>(xy), and 1>(x 2 ϩ y 2), because (14) Ϫy dx ϩ x dy y 2 x ϭ d a b, y Ϫy dx ϩ x dy xy x ϭ Ϫd aln b , y Ϫy dx ϩ x dy x ϩy 2 2 y ϭ d aarctan b . x How to Find Integrating Factors In simpler cases we may find integrating factors by inspection or perhaps after some trials, keeping (14) in mind.