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in 28. 29. 20 dx dy dx dy 2x - y 4x + 3y 2x + y x²-3y²

in 28. 29. 20 dx dy dx dy 2x - y 4x + 3y 2x + y x²-3y²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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27,29

4x be rewritten as x) - 4 (y/x) v-4 1- v' - 4 S. ariable v so that v = y/x, or ms of x, v, and dv/dx. ation (30) by the expressions /dx. Show that the resulting Lat V (29) ple. (30) (31) implicitly in terms of x. ) by replacing v by y/x in ome integral curves for and side of equation (29) c. This means that integral s on any given straight line The method outlined in Problem 25 can be used for any homogeneous equation. That is, the substitution y=xv(x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by y/x gives the solution to the original equation. In each of Problems 26 through 31: 26. 27. 28. 29. 30. 31. through the origin, although the slope changes from one line to another. Therefore, the direction field and the integral curves are symmetric with respect to the origin. Is this symmetry property evident from your plot? a. Show that the given equation is homogeneous. b. Solve the differential equation. G c. Draw a direction field and some integral curves. Are they symmetric with respect to the origin? adala ale ale ale dy dx dy dx dy dx dy dx dy dx dy dx 11 2.3 Modeling with First-Order Differential Equations 39 = = = x² + xy + y² x2 x² + 3y² 2xy E 4y - 3x 2x - y 4x + 3y !!! 2x + y x² - 3y² 2xy/ 10 Ing 3y2 - x2 mate 2xy Ledi to bedly co First-Order Differential S nmathematicians primarily because of the possibility y of problems in the physical, biological, and social matical models and their solutions lead to equations problem. These equations often enable you to make behave in various circumstances. It is often easy
Transcribed Image Text:4x be rewritten as x) - 4 (y/x) v-4 1- v' - 4 S. ariable v so that v = y/x, or ms of x, v, and dv/dx. ation (30) by the expressions /dx. Show that the resulting Lat V (29) ple. (30) (31) implicitly in terms of x. ) by replacing v by y/x in ome integral curves for and side of equation (29) c. This means that integral s on any given straight line The method outlined in Problem 25 can be used for any homogeneous equation. That is, the substitution y=xv(x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by y/x gives the solution to the original equation. In each of Problems 26 through 31: 26. 27. 28. 29. 30. 31. through the origin, although the slope changes from one line to another. Therefore, the direction field and the integral curves are symmetric with respect to the origin. Is this symmetry property evident from your plot? a. Show that the given equation is homogeneous. b. Solve the differential equation. G c. Draw a direction field and some integral curves. Are they symmetric with respect to the origin? adala ale ale ale dy dx dy dx dy dx dy dx dy dx dy dx 11 2.3 Modeling with First-Order Differential Equations 39 = = = x² + xy + y² x2 x² + 3y² 2xy E 4y - 3x 2x - y 4x + 3y !!! 2x + y x² - 3y² 2xy/ 10 Ing 3y2 - x2 mate 2xy Ledi to bedly co First-Order Differential S nmathematicians primarily because of the possibility y of problems in the physical, biological, and social matical models and their solutions lead to equations problem. These equations often enable you to make behave in various circumstances. It is often easy
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