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 +2Blocks 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 Units5. 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.
- Consider 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?b) Show that Δ2y0 = y2-2y1 + y0.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.Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.