naIs the solution unique?7. Consider the differential equationdrdt3kx - x³.a. If k≤ 0, show that the only critical value c = 0 of x is stable.b. If k > 0, show that the critical value c = 0 of x is now unstable, but that thecritical values c = ±√k are stable.c. Draw the bifurcation diagram-the plot of all points of the form (k, c) where cis a critical point of the differential equation, with arrows indicating stability andinstability. Note this will be a graph in the k-c plane.

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Answered: equation dx dt kx - x³. a. If k≤ 0,…

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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Consider the following system of differential equations describing two populations: æ' = 0.02a (1 – ) + 0.004xy 80 (1 y' = 0.04y(1 ) - 0.0002xy 20 a. Describe what the equations tell us about the populations (how the populations behave absent interaction, and how interaction affects each population). b. Find all equilibrium points algebraically. c. Classify each equilibrium using linearization d. Sketch some solution curves in the phase plane (x-y plane)
State whether the equation is ordinary or partial, linear or nonlinear and determine its order and degree.
mons One of the solutions of a linear homogeneous differential equation with constant coef- ficients is y = 2e-2x (2+x²cos (2x)). What is the minimum possible order of the differential equation? OA.7 OB. 5 O C. 2 O D. 9 O E. 1 O F. 10 O G.8 O H.3 OL. 11 OJ. 4 OK. 6 mmmuc wwminiw TEX Tamm Gum Tantimon Sinnitu Commamu umum
2. For a certain autonomous, first order differential equation with f as the dependent variable, you are told that there are 5 critical points (CP), located at -2 (an unstable CP), 0 (a stable CP), 2 (a semi-stable CP), 5 (an unstable CP), and 10 (a stable CP). a. Draw the appropriate Phase Portrait as a vertical line with critical points marked and arrows indicating direction of the rate of change. ( b. Sketch a graph of typical solution curve for values of y(0) above 10, between 10 and 5, between 5 and 2, between 2 and 0, between 0 and -2, and below -3 (including in the same sketch as your answer to part a.) t) c. If your initial value was f(0) = 15, what is the steady state solution for f d. What makes a Differential Equation autonomous?
11
Find the equilibrium points of = e -6y — 9 and classify each one as stable or unstable. dt Stable equilibria occur at y = Unstable equilibria occur at y = (If there is more than one equilibrium of a certain type, enter a comma-separated list. If there are no equilibria, enter "none".) On paper, sketch several solutions to the differential equation.
Determine the two singular points of the differential equation (x² - 81)y" + (9 − x)y′+ (x² + 18x +81)y = 0 List the points in increasing order: x1 = x2= Which of the following statements correctly describes the behaviour of the solutions of the differential equation near the singular point x1: A. All non-zero solutions are unbounded near 1. B. All solutions remain bounded near x1. C. At least one non-zero solution remains bounded near x1 and at least one solution is unbounded near x1. Which of the following statements correctly describes the behaviour of the solutions of the differential equation near the singular point 2: O A. All solutions remain bounded near 2. B. At least one non-zero solution remains bounded near 2 and at least one solution is unbounded near x2. C. All non-zero solutions are unbounded near 2.
(b) Consider the following phase line. y=2 y=-2 i. Write down a differential equation that has this phase line. ii. Describe the behaviour of the solution that satisfies y(-3) = 0 as the independent variable increases.
SOLVE THE FOLLOWING HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS. 1. (D³ - 5D² + 3D + 9)y = 0 2. (D³D²4)y = 0 3. (D² - 2D + 1)y = 0; when x = 0, y = 1, Dy = - 1. 4. (D6 + 8D4 + 16D²)y = 0 5. (D³+2D² - 5D - 6)y=0; y = 0, Dy = 0, D²y = 1, when x = 0.

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