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Practice Exam -3 Intro. DEs Spring 2005 1. Consider the folling system of differential equations which is supposed to model two interacting species: dx = 5x − x2 − xy, dt dy = −2y + xy dt (a) Would this system be modeling a coorperative, competetive, or predator-prey situation ?. Exlpain. (b) Find all equilibrium solutions. (c) Determine the type and stability of the critical points. 1 2 2. Assume that the following system is almost linear: dy dx = f (x, y), = g(x, y) dt dt Let (0, 0) be a critical point of the system. Prove dx dy = fx (0, 0)x + fy (0, 0)y, = gx (0, 0)x + gy (0, 0)y dt dt is the linearized system of the almost linear system at (0, 0). 3 3. Consider the linear system dy dx = ǫx − y, = x + ǫy dt dt (a) Determine the type and stability of the critical point (0, 0) if (i) ǫ < 0 (ii) ǫ = 0 (iii) ǫ > 0 (b) Explain why small perturbation of the system change the type and stability of the critical point. 4 4. Let f (t) be continuously differentiable function such that |f (t)| ≤ Mect ∀t. Denote F = L{f (t)}. Prove the followings: (a) L{f ′ (t)} = sF (s) − f (0) for any s > c. (b) L{eat f (t)} = F (s − a) for any s > a + c. 5 5. Using Laplace Transform Method, solve the initial value problem (a) x′′ + 4x = cos 3t, x′ (0) = x(0) = 0. (b) y ′′ + 4y ′ + 4y = 8t2 , y(0) = y ′ (0) = 0 6 6. Using Laplace Transform Method, find the solution of the system x′′ = −3x + 2y, y ′′ = 2x − 2y + 40 cos 3t subject to the initial conditions x(0) = x′ (0) = y(0) = y ′ (0) = 0. 7 L{tn } = n! , s>0 sn+1 1 L{eat } = , s > a, s−a s L{cos kt} = 2 , s>0 s + k2 k , s>0 L{sin kt} = 2 s + k2 s L{cosh kt} = 2 , s > |k| s − k2 k L{sinh kt} = 2 , s > |k| s − k2