Consider the plant equation
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Differential Equations: An Introduction to Modern Methods and Applications
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- Which of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system yi = -(y1 – y2)e(un -)* Ý2 = (y1 – y2)e(% 4)* – 2y? %3D Select one: a. V(y1, Y2) = e(y-4)² O b. V(y1, Y2) = y5 O c. V(y1, Y2) = -e-92)* + y – 1 + y – 1 d. V(y1, Y2) = elon 2)² * – y Y2arrow_forward5. Consider the following nonlinear predator-prey model: = 6y₁2yY1Y2, dy₁ dt dy2 dt where = y1y2 - 2y2, where y₁ (t) is the population of the prey species, and y2(t) is the population of the predator species. We can write this in vector form as y' = F(y), 6y₁-2y1-9192 F(y) = - (5 (37-302)) = (0934-201² - 31.12). (91, 22 (a) Find the steady states of this system, i.e. the vectors (y1, 92) so that F(y₁, y2) = (0,0). Hint: there are three steady states. (b) Calculate the total derivative DF (y1, y2) = əfi Əf₁ дуг Əy₁ af₂ Əf2 дуı дуг as a function of y₁ and y2. (c) For each steady state (y1, 92) from (a), find the eigenvalues of DF(y1, 92) and classify the steady state as stable or unstable.arrow_forwardThe Lotka-Volterra equations are often used to model the links between a particular population of prey organisms and a population of predatory organisms. In a particular ecosystem u is used to represent the number of predatory organisms and v to represent the number of prey organisms. Suppose the growth rate uv of the predatory organisms is f(u,v) = - 0.5u + and of the prey organisms is g(u,v) = 6v – 10uv. 100 (a) Show that if u = 0.6 and v = 50, then f (u,v) = 0, and g(u,v) = 0. (The populations are said to be in equilibrium.) %D %3D f(u,v) (b) Find the linear approximation of the vector valued function h:(u,v)→ if u is close to 0.6 and v is g(u,v) close to 50. (a) Evaluate f(u,v) at u = 0.6 and v = 50, f(0.6, 50) = (Type an integer.)arrow_forward
- Consider the linear system y' = (a) Find the general solution. Ay with 4 - ( -5₁ 4 ) Aarrow_forwardIf yı and Y2 are linearly independent solutions of t²y" + 4y' + (5 + t)y = 0 and if W(y₁, y₂)(1) = 5, find W(y1, y2)(4). Round your answer to two decimal places. W(y1, y2)(4) = iarrow_forwardObtain characteristics for y2 Uxx + Uyy = %3Darrow_forward
- This is the first part of a three-part problem. Consider the system of differential equations y1 - 2y2, y1 + 4y2. Yi Y2 Rewrite the equations in vector form as '(t) = Ay(t). j'(t) = 身arrow_forwardFind the equation of the family of orthogonal trajectories of the system of parabolas y? = 2x + C. y = Cex y = Ce2x y = Cex y = Ce 2xarrow_forwarda.) Form the complimentary solution to the homogeneous equation. y_c(t) = c_1 [ _ _] + c_2 [ _ _] b.) Construct a particular solution by assuming the form y_P(t) = e^(-3t)a and solving for the undetermined constant vector a. y_P(t) = [ _ _] c.) Form the general solution y(t) = y_c(t) + y_P(t) and impose the initial condition to obtain the solution of the initial value problem. [y_1(t) y_2(t)] = [ _ _]arrow_forward
- Solve the system Ÿ' = AŸ + Ġ for Aarrow_forward13. Find the equation of the orthogonal trajectories of the system of parabolas y² = 2x + 2. a. y = ce-x b. y = cex c. y = ce²x d. y = ce-2xarrow_forwardA chemistry student heats a beaker that is at room temperature (30 ° C)by inserting it in an oven that has been preheated to 250 °C. If u is the temperature of the beaker in °C, what is the appropriate equation to model the temperature of the beaker. O du/dt = k(u –- 250) O du/dt = k(u – 30) O du/dt = k(250 – u) O du/dt = k(30 – u)arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,