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In Problems 13–32 use variation of parameters to solve the given nonhomogeneous system.
17.
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- Q.8arrow_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_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
- Tr.28.arrow_forward8.3 I only need number 14 pleasearrow_forward1.1 A mathematical model that describes a wide variety of physical nonlinear systems is the nth-order differential equation y(n) = 9 (t, y, y, %3D .... where u and y are scalar variables. With u as input and y as output, find a state model.arrow_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_forwardSolve the given differential equationsarrow_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
- .6. Find the solution of the differential system x' = ( )x+(). 1 4 d ), #(0) = 0, y(0) = 1, %3D -1 -2t e dt -3arrow_forwardPROBLEM 2.2 Solve the system of ODEs for t = [0,50] with 1.C.: x(0) = -8, y(0) = 8, z(0) = 27. 'x' (t) = -10x+10y y' (t) = 28x - y - xz 8 z' (t) = -z + xyarrow_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_forward
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