Problem 7.6 Let S = {(ao, a1, a2, . . )} be the set of all sequences ofreal numbers, and let P = {Eo Cnr"} be the set of all power seriesin r with real coefficients. Define the function y : S → P by%3Dy(ao, a1, a2, . . .) =1".n!n20(a) Let k E R. Show that y((1, k, k², k³, . )) is the expontial func-tion ekz.(b) Find all solutions to the recurrence relation a, = ka, 1-(c) Find all solutions to the differential equation y'(t) = ky(t).(d) How are these two sets of solutions related by y ?

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Answered: r(a0, a1, a2, . . ) =) n! n20 (a) Let k…

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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The solution of the differential equation (xy – y²)dx – (x² + y²)dy = 0 is : %3D A) 2 In + 1| – In | - = - Inx + c B) In + 1| – in – in|| = - In 2| -:--Inx + c = -Inx + c |3D y -Inx + C %3D C) In ( + 1)"* – In | D) 2 In – 1|– in + 2|– In|| = -Inx + c E) In + 2| – tn |2| - - Inx + c %3D y %3D y
Given the differential equation: Choose all correct answers. A (x²-1) y" - 6x y¹ + 12y=0 D The recurrence relation is given by: (n-2)(n-3) C C n+2 (n+2) (n+3) n y=c (1+6x4+x³) +c₂(x+x² + x³) (C) y=c (1+6x²+x4) +c₁(x+x³) = The recurrence relation is given by: C n+2 = (n-3)(n-4) (n+1) (n + 2) C n n = 0,1,2,... n=0,1,2,...
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D
Interaction of two species of squirrels fiercely competing for the same ecological niche on an island is described by Lotka-Volterra-Gause equations dN1 = N1(2 – N1 – 2N2) = f(N1, N2), dt (1) dN2 N2(3 – N2 – 3N) = g(N1, N2), dt where N = N1(t) and N2 = N2(t) are the population densities of the competing species. (a) Find all equilibria in model (1) and explain biological meaning of each of them. Which of the equilibria are biologically feasible? (b) For each of the equilibria write a Jacobian matrix and find its eigenvalues. Classify all equilibria points.
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r(a0, a1, a2, . . ) =) n! n20 (a) Let k E R. Show that y((1, k, k² , k³, . )) is the expontial func- tion ekr. (b) Find all solutions to the recurrence relation a, = ka, 1. (c) Find all solutions to the differential equation y'(t) = ky(t). (d) How are these two sets of solutions related by y?