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In Problems 13–32 use variation of parameters to solve the given nonhomogeneous system.
21.
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Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's ... Problems, 9th Edition, Single-Term
- 12. [Kaplan & Glass(1995)] Limpets and seaweed live in a tide pool. The dynamics of this system are given by the differential equations ds s² – sl, = S dt dl sl --2, 1>0,s > 0, dt 2 where the densities of seaweed and limpets are given by s and l, respectively. (i) Determine all equilibria of this system. (ii) For each nonzero equilibrium determined in part (a), evaluate the stability and classify it as a node, focus, or saddle point. (iii) Sketch the flows in the phase plane. (iv) What will the dynamics be in the limit as t → o for initial conditions (i) s(0) = 0, 1(0) = 0? (iї) s(0) — 0, 1(0) — 15? (iii) s(0) = 2, 1(0) = 0? (iv) s(0) = 2, 1(0) = 15?arrow_forwardProblem 2. Consider the equation: x?y"(x) – xy' +y = 0. Given that yı(x) = x is a solution of this equation. Use the method of reduction of order, find the second solution y2(x) of the equation so that y1 and y2 are linearly independent. (Hint: y2(x) should be given in the form y2(x) = u(x)y1(x). Substitute it into the equation to find u(x).) %3Darrow_forward8.3 I only need number 14 pleasearrow_forward
- 1. The Lotka-Volterra or predator-prey equations dU = aU – UV, dt (1) AP = eyUV – BV. dt (2) have two fixed points (U., V.) = (0,0), (U., V.) = (- :). The trivial fixed point (0,0) is unstable since the prey population grows exponentially if it is initially small. Investigate the stability of the second fixed point (U..V.) = 6:27 PM 3/3/2021 近arrow_forward3. Suppose ay" + by' + cy = 0 with y(0) = d and y'(0) = k has a general solution y 4e2 - What are the constants a, b, c, d, and k ?arrow_forwardSolve the given differential equationsarrow_forward
- Use (1) in Section 8.4 X = eAtc (1) to find the general solution of the given system. 1 X' = 0. X(t) =arrow_forward7. Find the general solution of the given system of equations. (a) (b) y' - [ = -1 3 -2 2 [ y + ex X 2-5 0 y = [² =2] y + [cz] · 1 -2 0 < x < T.arrow_forwardConsider the system = (41%) (22) + (1) * น in which a is a constant (a) Determine the condition under which the system is controllable (b) For a 1 (i) Show that et4 = (cos(t)) sin(t)) cos(t)) (Hint: You may note that A4 = 1, A4k+1 = A, A4k+2 = −I, A4+3 = -A for all k > 0 and determine the MacLaurin series expansion of cos(t) and sin(t)) To (ii) Write the integral formula for the solution X (t) in terms of X (0) = X₁ = and u. Yo (ii) Extract a separate formula for each component of X(t) = ((0)arrow_forward
- 19. Find the general solution of the system of differential equations (2)-(; 3)(;) Sketch the phase portrait. Classify the origin and determine its stability.arrow_forwardProblem 2. Let dX/dt = AX. Find et and use it to write down the solution to the system. (²2) b) 4-(32) 1 a) A = Aarrow_forwardTr.28.arrow_forward
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