Find the breaking time t, and breaking position x, for the following nonlinear wave problem ų +(1+2u)uz = 0; u(x,0) = e¯.
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- Q.4 Solve the wave equation: U =U +t+1- 1) =5, U(7,t) = cost U(x,0) = , U,(x,0) =Which of the following functions V(y1, Y2) is a Lyapunov function for the dynamical system yi = -(y1 – y2)e(un -)* Ý2 = (y1 – y2)e(% 4)* – 2y? %3D Select one: a. V(y1, Y2) = e(y-4)² O b. V(y1, Y2) = y5 O c. V(y1, Y2) = -e-92)* + y – 1 + y – 1 d. V(y1, Y2) = elon 2)² * – y Y211.6: Problem 8 Find the directional derivative of f(x, y, z) point (1, 2, 3) in the direction of a vector making an angle of 2 with V f(1, 2, 3). = zx + y“ at the
- 104v 8. 2 -10 -8 -6 -4 -2 0 4 6 8 10 -2 -4 -6 -8 -10 Match the graph with its function by translating the graph of y = y = Vx+2 y = Vx-2 y = Vx-2 Py = Vx +2Consider the order and linearity of the following equations: i. (y+t)dy/dt + y=1 ii. 3dy/dt + ( t+4)y = t2+ d2y/dt2 iii. d2y/dt2 = cos(2ty) Which of the equations is/are of order two and non-linear?Blocks A (mass 5.00 kg) and B (mass 6.50 kg) move on a frictionless, horizontal surface. Initially, block B is at rest and block A is moving toward it at 5.00 m/s. The blocks are equipped with ideal spring bumpers. The collision is head-on, so all motion before and after the collision is along a straight line. Let +x be the direction of the initial motion of block А. Part C Find the velocity of block B when the energy stored in the spring bumpers is maximum. Express your answer with the appropriate units. HÀ VB = Value Units Submit Request Answer Part D Find the velocity of block A after they have moved apart. Express your answer with the appropriate units. HÅ ? VĀ = Value Units
- b) Show that Δ2y0 = y2-2y1 + y0.5. Consider the following nonlinear predator-prey model: = 6y₁2yY1Y2, dy₁ dt dy2 dt where = y1y2 - 2y2, where y₁ (t) is the population of the prey species, and y2(t) is the population of the predator species. We can write this in vector form as y' = F(y), 6y₁-2y1-9192 F(y) = - (5 (37-302)) = (0934-201² - 31.12). (91, 22 (a) Find the steady states of this system, i.e. the vectors (y1, 92) so that F(y₁, y2) = (0,0). Hint: there are three steady states. (b) Calculate the total derivative DF (y1, y2) = əfi Əf₁ дуг Əy₁ af₂ Əf2 дуı дуг as a function of y₁ and y2. (c) For each steady state (y1, 92) from (a), find the eigenvalues of DF(y1, 92) and classify the steady state as stable or unstable.Reproduce the given computer-generated direction field, Then sketch an approximate solution curve that passes through each of the Indicated points. Y-1- xy. xp 2. -2 2. (a) y(0) - 0 (b) y(-1) = 0 (C) y(2)=2 ) y (0) : -4
- Find the orthogonal trajectory of y=c cos x4.2. For the regression model y, =Bo+& with n=2 and y'=(2, 4), draw the data in two-dimensional space. Identify the orthogonal projection of y onto L(X)=L(1). Explain geometrically Bo, û, and e.The survival of two species depends on their mutual cooperation. For instance, a species of bee that feeds primarily on the nectar of one plant species and simultaneously pollinates the plant. One simple model of mutualism is given by dx/xt = −ax + bxydy/dt = −my + nxy where a, b, m, and n are all positive constants. Perform a graphical analysis and indicate the trajectory directions in the phase plane.