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In Problems 13–32 use variation of parameters to solve the given nonhomogeneous system.
18.
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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
- .1) The general solution of the equation y" +y =-5e-* , is: o y(x) = Ce-x + ez (Acosx + Bsinx) -- O y(x) = Ce-* + ez (Acosx + Bsinx) -xe-x Oy(x) = Ce-* + ez(Acosx + Bsinx) +xe-* %3D ) y(x) = Ce-* + ez (Acosx + Bsinx) +e-* %3Darrow_forwardConsider the following initial value problem: Edit 1 y" + 8y + 15y = 8(t – 5) + u10(t); y(0) = 0, y(0) = = 4 a) Find the solution y(t). ("-") 1 (e-St – e 5) y(t) = 8. 1 -3(t-5) 1 –5(t–5) ult) X 2 1 e-5(t-10) 10 1 -3(t-10) 6. Ud(t) 15 where c = 5 and d 10arrow_forward1. 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_forward
- 7. A scientist places two strains of bacteria, X and Y, in a petri dish. Initially, there are 400 of X and 500 of Y. The two bacteria compete for food and space but do not feed on each other. If x = x(t) and y = y(t) are the numbers of strains at time t days, the growth rates of the two populations are given by the system x' = 1.2x – 0.2y, y' = -0.2x + 1.5y Determine what happens to these two populations by solving the system of differential equations.arrow_forward12. [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_forward9. discuss the behavior of the dynamical system Xk+1= Axk where -0.31 (@) A = [ (b) A = ["0.3 11 1.5 0.3 (b) A = 0.3 1 3Darrow_forward
- 8.3 I only need number 14 pleasearrow_forwardQ. No. 11 The solution of the DE 3ry" + y/ – y = 0 (a) yı = rš[1 – {x +²+...], y2 = 1+x – 20² + ... (b) yı = a3[1 – r +a² + ...], y2 = 1+ 2x – 2x² + ... (c) yı = xš[1 – x + a² + ...], y2 =1+ 2x – 2x3 + ... (d) yı = [1 – x + x² + ...], y2 = 1+ 2x – 2x2 +... solve this and tick the correct optionarrow_forwardProblem 11.3. Find a solution to the equation Utt = Uxx, x ≤ [0; 1], t ≥ 0, with ut(x,0) = 0, for x = [0; 1/2], u(0, t) = 0, u(1, t) = 0, u(x, 0) = 0, ut(x,0) = 1, for x € [1/2; 1],arrow_forward
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