18. Convert the second-order equation d²y 0 dt2 into a first-order system using v = dy/dt as usual. (a) Find the general solution for the dv/dt equation. (b) Substitute this solution into the dy/dt equation, and find the general solution of the system. (c) Sketch the phase portrait of the system.

a) Find the initial value problem of the system of linear differential equations: [1 -3 x'(t) = 3 х, х(0) — 1 b) Describe the behavior of the solutions as t -→ 0.

Find all the equilibrium points for the system below. At each equilibrium point, linearize the system and classify whether the equilibrium is small signal stable or unstable by computing the equilibrium points at each equilibrium dx -x+0.5x,*x2 dt dx 2 = -x,+0.5x,*x, dt

3t Verify that the pair x(t) = ³t, y(t) = ¹ is a solution to the given system below. Sketch the trajectory of the given solution in the phase plane. e dx dt Ay 3 dy dt Verify that the pair x(t), y(t) is a solution of the system. Select the correct choice below and fill in any answer boxes to complete your choice. (Type expressions using t as the variable.) O A. 3t The pair x(t) = ³t, y(t) = ¹ is a solution because in terms of t. x(t) dt = | 10- =3y³. and i Sy(t) dt = | and y' (t) = OB. The pair x(t) = e³t, y(t)=e¹ is a solution because in terms of t, x'(t)= 3t Sketch the trajectory of the given solution in the phase plane. Choose the correct graph below. O A. 0- =y 100 O D. A) 10+ H 0 10 Q Q Q Q ✔ In terms of x and y, x'(t) = O B. Ay 100- O E. Ay In terms of x and y, x(t) dt = and y(t) dt = | || and y'(t)= 100+ 10 10 Q Q Ⓡ O C. A 10- 0 O F. Ay 10+ 0- 10 10 Q Q ✔ Q