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rin 29. dy dx 4.x + 3y 2x + y

rin 29. dy dx 4.x + 3y 2x + y

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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N 25. Consider the equation mortion besesnqxo nadw vinsup dy y - 4x misoliamsdam a onsmedie (29) dx x - y 2200010 a. Show that equation (29) can be rewritten as = dy dx or = (y/x) - 4 1 − (y/x) ' thus equation (29) is homogeneous. V y/x, or b. Introduce a new dependent variable y so that y = y = xv(x). Express dy/dx in terms of x, v, and dv/dx. c. Replace y and dy/dx in equation (30) by the expressions fordib. from part b that involve v and dv/dx. Show that the resulting differential equation is dv v+x- = dx dv - x dx = v-4 1 - v' ²-4 1-v siquiae The method outlined in Problem 25 can be used for any to homogeneous equation. That is, the substitution y = xv(x) transforms (30) 0) 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: (31) Observe that equation (31) is separable. 020 1,0 d. Solve equation (31), obtaining v implicitly in terms of x. e. Find the solution of equation (29) by replacing v by y/x in the solution in part d. f. Draw a direction field and some integral curves for equation (29). Recall that the right-hand side of equation (29) actually depends only on the ratio y/x. This means that integral curves have the same slope at all points on any given straight line 26. 27. 28. 29. 30. 10007 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? 31. a. Show that the given equation is homogeneous. Solve the differential equation. Gc. Draw a direction field and some integral curves. Are they symmetric with respect to the origin? x² + xy + y² x2 x² + 3y² a ada als als ala dy dx dy dx dy dx dy dx dy dx dy dx 2.3 Modeling with First-Order Differential Equations 39 2xy 4y - 3x 2x - y 4x + 3y 2x + y x² - 3y² 2xy x3y²-x² 2xy = || = ballquesto ba
Transcribed Image Text:N 25. Consider the equation mortion besesnqxo nadw vinsup dy y - 4x misoliamsdam a onsmedie (29) dx x - y 2200010 a. Show that equation (29) can be rewritten as = dy dx or = (y/x) - 4 1 − (y/x) ' thus equation (29) is homogeneous. V y/x, or b. Introduce a new dependent variable y so that y = y = xv(x). Express dy/dx in terms of x, v, and dv/dx. c. Replace y and dy/dx in equation (30) by the expressions fordib. from part b that involve v and dv/dx. Show that the resulting differential equation is dv v+x- = dx dv - x dx = v-4 1 - v' ²-4 1-v siquiae The method outlined in Problem 25 can be used for any to homogeneous equation. That is, the substitution y = xv(x) transforms (30) 0) 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: (31) Observe that equation (31) is separable. 020 1,0 d. Solve equation (31), obtaining v implicitly in terms of x. e. Find the solution of equation (29) by replacing v by y/x in the solution in part d. f. Draw a direction field and some integral curves for equation (29). Recall that the right-hand side of equation (29) actually depends only on the ratio y/x. This means that integral curves have the same slope at all points on any given straight line 26. 27. 28. 29. 30. 10007 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? 31. a. Show that the given equation is homogeneous. Solve the differential equation. Gc. Draw a direction field and some integral curves. Are they symmetric with respect to the origin? x² + xy + y² x2 x² + 3y² a ada als als ala dy dx dy dx dy dx dy dx dy dx dy dx 2.3 Modeling with First-Order Differential Equations 39 2xy 4y - 3x 2x - y 4x + 3y 2x + y x² - 3y² 2xy x3y²-x² 2xy = || = ballquesto ba
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