Solve the initial value problem x'(t)=Ax(t) for t20, with x(0)=(5,4). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by x'=Ax. Find the directions of greatest attraction and/or repulsion. A= 11 -1 3 7 Solve the initial value problem. x(t)= ...

Which of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system y1 = -2y1yževí v½ – Y1, ý2 = -2yjyzeyi vi Select one: O a. V(y1, Y2) = e1 ½ + O b. V(y1, Y2) = evi v? + }yi – 1 O c. V(y1, Y2) =-y1y2eiv½ – y? O d. V(y1, 42) = -evi v? +

Which of the following functions V(y₁, y2) is a Lyapunov function for the dynamical system with equilibrium point at (0, 0) y₁ = −2y₁y¾e(Y192)² – 6y₁, y₂ = -2y²y₂e(³₁V2)² – 2y₂ Select one: a. ○ b. V(y₁, y2) = e(Y1Y2)² V(y₁, y2) = e(¹¹³₂)² + y² −1+3y² ○ c. V(y₁, ₂) = y²e(₁8₂)² + y² − 3y₁ O ○ d. V(y₁, y₂) = y† + (y₂ − 1)²

5. Determine whether the functions y, and y2 are linearly dependent on the interval (0,1). Yı = 1+ e', Y2 = 1 – 2e2t

= 1. A particle 1 of mass m₁ is attached to one end of a spring with constant k at position x = = 0. It is initially at rest, with velocity u₁ 0. The other end of the spring is attached to a fixed wall at position x = -l where l is the length of the spring at rest. The whole setup is horizontal along the x-axis. Another particle, 2, of mass m2 travels towards particle 1 with constant velocity u2 (note that u2 < 0 in this setup). The two particles undergo a collision with coefficient of restitution 0 ≤ e ≤ 1. (a) Using the one-dimensional equations of collision (eq (6.44) in the lecture notes) show that immediately after collision the velocity of particle 1 is given by V₁ = (1 + e) m2u2 m1 + m2 Find the velocity v2 of particle 2 immediately after collision. What happens to particle 2 if m₁ = m2 and the collision is elastic (e = 1)? (b) Immediately after collision, particle 1 has velocity v₁ and is at position x = 0, which we take as initial conditions for its subsequent motion under the…

10. Determine three linearly independent solutions to the equation y" + 2y" – 3y = 0 of the form y(x) = e"*, where r is a real number. Remember to prove that these solutions are indeed linearly independent.

Draw the slope fields of dy/dt = -t2y at integer values ( y is greater than or equal to -5 and less than or equal to 5 and x is greater than or equal to -5 and less than or equal to 5). Sketch the solution curves with initial values y(0) = 2 y(0) = -2 y(-4) = 3 y(-4) = -3

solve b

Which of the following functions V(₁, ₂) is a Lyapunov function for the dynamical system with equilibrium point at (0, 0) V₁ = −2y₁³e(K)² – 6₁₁₁=²²₂)² – 2y₁ Select one: O a. V(V₁, V₂) = e(₂)² V(₁/₂) = ³₂)² + y² −1+3y² Ob. Oc. V(₁/₂) = ₁ + (Y₂ − 1)² O d. V(₁₁/2) = ye()² + y² - 3₁

please help show the step on how to get the correct asnwer. i have attached the correct answer   Question: solve the given initial-value problem.sketch the graphof both the forcing function and the solution. y″ + y′ +3y = u(t –2)y(0)= 0y′(0) = 1