In each of Problems 1 through 26:(a) Find the general solution in terms of real functions.(b) From the roots of the characteristic equation, determinewhether each critical point of the corresponding dynamicalsystem is asymptotically stable, stable, or unstable, and clas-sify it as to type.(c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x = x₁i + x₂j = yi+y'j ofthe corresponding dynamical system. Then sketch by hand,or use a computer, to draw a phase portrait, including anystraight-line orbits, from this family of trajectories.1. y" + 2y' - 3y = 0

Given Information:The differential equation is .Objective:a) Find the general solution.b) Analyze…

Answered: In each of Problems 1 through 26: (a)…

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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Thus far in the class we've only seen one kind of so-called "linear" difference equation models: the Malthusian ones of the form N(t+1)= XN (t) This is called linear because N(t+1) is a linear function of N(t) - in other words, if we view N(t+1) as "y" and N(t) as "r" then this is an equation of the form "y = mx" since À is a constant. Another more general type of "linear" difference equation is: N(t+1) = AN(t) + a where a is some constant. This is again a linear model because N(t+1) is a linear function of N(t), this time of the form "y=mx+b". This is sometimes called a "linear difference equation with a constant term". The next few problems have to do with linear difference equations with constant terms. 4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 *…
(a) Suppose we create a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2). Height is measured in cm, bacteria is measured in thousand per ml of soil (so 1000 is 1 unit), and type of sun = sun and type of sun = between shrubs with a 5500 bacteria and those with a 4000 bacteria count. Assuming the sun is the 0 if the plant is in partial 1 if the plant is in full sun. What will be the height difference on average same for both. (b) Let's say after a few training steps your model is in this state: Y = 33 + 4.6 * X1+8* X2. Explain what the effect of the plant being in sun vs. partial shade will have on the average height of the shrub if there are no bacteria in the soil, based on the model.
In each of the following problems [8]-[9]: [8] [9] (a) find all equilibrium solutions of the equation (*); (b) for each equilibrium point, write down the linear approximating equation near the equilibrium and determine whether the equilibrium is stable, asymptotically stable or unstable with respect to the linear approximating equation; (c) try to use the linear stability/instability obtained in (b) to determine whether each of the equilibria is stable, asymptotically stable or unstable with respect to the nonlinear equation (*); (d) if the linear approximation obtained in (b) was not enough to determine the stability of an equilibrium with respect to the nonlinear equation (*), use other methods to determine whether the equilibrium is stable, asymptotically stable or unstable with respect to the nonlinear equation (*) (*) (*) dy dt dy dt = = y(y − 1)(y + 2) = y(y − 1)(y + 2)² -
Can someone please show me step by step how to solve this exercise
A salmon population is modelled by the DTDS X+1 = 0.3 + x, • (a) ( Write down the updating function f (x) of this DTDS. (b) ( Suppose we know that X10 = 0. 5. What is the value of X11? • (c) SWrite down the equation that we need to solve in order to find the fixed points. Then solve the equation algebraically (that is, without using graphs). • (d) point. (No marks will be given if you use other methods.) Use the Stability Theorem to determine the stability of each fixed
The same disease is spreading through two populations, say Pi and P2, with the same size. You may assume that the spread of the disease is well described by the SIR model. dSa - BaSaIa dt dla - dt dRa YaIa dt with Sa(0) + I.(0) + R.(0) = N where N denotes the fixed population size. The subscript a identifies the population P or P2. For example, if a = 1, the variables are related to P1. Assume that S1 (0) = S2 (0) and I1(0) = I½(0) and that no interventions such as quarantine or vaccination have been implemented. If the difference in the spread of the disease is due only to the poor over-all health of a population, which population has the best over-all health of the two populations? - Susceptible Population 1 - Susceptible Population 2 700 600 500 400 300 200 100 10 15 20 t P1. P2. Susceptible
VI. 1 Only
A firm has the following production function:q = ƒ (z1, z2) = z1αz2βb. Solve for the firm's conditional demand function (simplify)

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