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25. y' = ey- oy³, e > 0 and o > 0. This equation occurs in the study of the stability of fluid flow.

25. y' = ey- oy³, e > 0 and o > 0. This equation occurs in the study of the stability of fluid flow.

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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Question

25

 

✓ constant, Ifc = -1, ₁ (t) is he second and that ue of the rt > 0, Sc=0 ile that O, then (27) (28) (1) 9) 1-n 23. a. Solve Bernoulli's equation when n = 0; when n = 1. b. Show that if n 0, 1, then the substitution y = y¹-" reduces Bernoulli's equation to a linear equation. This method of solution was formulated by Leibniz in 1696. In each of Problems 24 through 25, the given equation is a Bernoulli equation. In each case solve it by using the substitution mentioned in Problem 23b. 24. y' = ry -ky², r> 0 and k > 0. This equation is important in population dynamics and is discussed in detail in Section 2.5. where 25. y' = ey- oy³, e > 0 and o> 0. This equation occurs in the study of the stability of fluid flow. Discontinuous Coefficients. Linear differential equations sometimes occur in which one or both of the functions p and g have jump discontinuities. If to is such a point of discontinuity, then it is necessary to solve the equation separately for t < to and t > to. Afterward, the two solutions are matched so that y is continuous at to; this is accomplished by a proper choice of the arbitrary constants. The following two problems illustrate this situation. Note in each case that it is impossible also to make y' continuous at to. 26. Solve the initial value problem where y' + 2y = g(t), y(0) = 0, Dus Diff g(t) = = 27. Solve the initial value problem (1, 0≤t≤ 1, t > 1. y + p(t) y = 0, y(0) = 1, p(t) = >= {². di 2, 0≤ t ≤ 1, 109 nomulos VORHER thblugnia odi amaidon t > 1. 42926
Transcribed Image Text:✓ constant, Ifc = -1, ₁ (t) is he second and that ue of the rt > 0, Sc=0 ile that O, then (27) (28) (1) 9) 1-n 23. a. Solve Bernoulli's equation when n = 0; when n = 1. b. Show that if n 0, 1, then the substitution y = y¹-" reduces Bernoulli's equation to a linear equation. This method of solution was formulated by Leibniz in 1696. In each of Problems 24 through 25, the given equation is a Bernoulli equation. In each case solve it by using the substitution mentioned in Problem 23b. 24. y' = ry -ky², r> 0 and k > 0. This equation is important in population dynamics and is discussed in detail in Section 2.5. where 25. y' = ey- oy³, e > 0 and o> 0. This equation occurs in the study of the stability of fluid flow. Discontinuous Coefficients. Linear differential equations sometimes occur in which one or both of the functions p and g have jump discontinuities. If to is such a point of discontinuity, then it is necessary to solve the equation separately for t < to and t > to. Afterward, the two solutions are matched so that y is continuous at to; this is accomplished by a proper choice of the arbitrary constants. The following two problems illustrate this situation. Note in each case that it is impossible also to make y' continuous at to. 26. Solve the initial value problem where y' + 2y = g(t), y(0) = 0, Dus Diff g(t) = = 27. Solve the initial value problem (1, 0≤t≤ 1, t > 1. y + p(t) y = 0, y(0) = 1, p(t) = >= {². di 2, 0≤ t ≤ 1, 109 nomulos VORHER thblugnia odi amaidon t > 1. 42926
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