5. Use the separation of variables method to solve the heat equation Ut = 0 < x < 2n, t > 0 u(0, t) = u(27, t) Uz(0, t) = uz(27, t) u(x, 0) = f(x) t > 0 %3D t > 0 0
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- Consider underdamped forced system governed by y''+y'+3y = 4cos(3t). Find the general solution of this nonhomogeneous equation when y(0) = 6, y'(0) = 0.2. Factory A produces a solution of 40% concentration of chemical Y at a rate of 7 kg/min, while factory B produces a solution of 20% concentration of chemical Y at a rate of 4 kg/min. Both solutions are fed into a mixer containing 60 kg of a 25% concentration solution of chemical Y. The mixture leaves the mixer at a rate of 10 kg/min. Assuming uniform mixing, what will be the concentration of chemical Y in the final solution after 20 minutes?2)A multiunit mixing process consists of two tanks T1 and T2. Ten (10) gal/min of pure water is pumped from the outside into Tank 1 which initially contains 100 gal of water in which 120 Ib of salt are dissolved. Tank 2 initially contains 200 gal of water in which 50 Ib of salt are dissolved. Liquid is circulated continuously between the two tanks by pumping 12 gal/min from Tank 1 into Tank 2 and pumping 2 gal/min from Tank 2 into Tank 1. Also, 10 gal/min of the uniform solution in Tank 2 is taken out of the system. Thus, the volume of the content of each tank remains constant and the contents of the two tanks are kept uniform by stirring. Find the design equations that would be necessary to calculate the salt contents y1 (t) and y2 (t) in Tank T1 and Tank T2 respectively at any time t. What would be the salt contents in the tanks after a long time? When will the salt contents of the two tanks be equal and what will be the salt contents in the tanks at that time? Calculate your…
- Use any numerical method to solve the given coupled equation system: y′ + 2xz = x2, y(0) = 1 z′′′ − y2z′ + z = 0, z(0) = 1, z′(0) = 0, z′′(0) = −1 Find the solution for y(x) and z(x) at x = 1 and x = 212.Consider the system dx/dt=x(a−σx−αy),dy/dt=y(−c+γx),dx/dt=xa−σx−αy,dy/dt=y−c+γx, where a, σ, α, c, and γ are positive constants. a.Find all critical points of the given system. How does their location change as σ increases from zero? Assume that a/σ > c/γ, that is, σ < aγ/c. Why is this assumption necessary? b.Determine the nature and stability characteristics of each critical point. c.Show that there is a value of σ between zero and aγ/c where the critical point in the interior of the first quadrant changes from a spiral point to a node. d.Describe the effect on the two populations as σ increases from zero to aγ/c.Consider the equation ut=uxx−9u, 0<x<1, t>0, with boundary condition u(0,t)=0,u(1,t)=4sinh(9), and initial condition u(x,0)=4sinh(9x)+5sin(3πx). We wish to solve. You have to do this in two steps, first make the boundary conditions homogeneous, then get rid of the heat loss term. Perhaps useful is to first solve for the steady state u-steady state(x) = ? Then the final solution is u(x,t) = ?
- 16.Consider the autonomous system dxdt=y,dydt=x+2x3.dxdt=y,dydt=x+2x3. a.Show that the critical point (0, 0) is a saddle point. b.Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x → 0, y → 0 as t → ∞ is y = −x. c.Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = −x and y = x for the linear system.consider the nonlinear second-order ODE x''+2x'+4x/(1+x^2)=0 a. convert to DE to a system b. Find all critical points of the resulting systemYou are given a pair of differential equations: (dx/dt) = 1 - y, (dy/dt) = x2 - y2 1. Find all of the equilibrium of these equations 2. Classify each equilibrium point of this non-linear system as far as possible by considering the Jacobian matrix
- 3. i) Determine the general solution of the system of equations indicating the method applied. ii) The particular solution by applying the initial conditions: ..6.By introducing suitable dimensionless variables, we can write the system of nonlinear equations for the damped pendulum (Equations (8) of Section 9.3) as dxdt=y,dydt=−y−sin x.dxdt=y,dydt=−y−sin x. a.Show that the origin is a critical point. b.Show that although V(x, y) = x2 + y2 is positive definite, ⋅V(x,y)V⋅x,y takes on both positive and negative values in any domain containing the origin, so that V is not a Liapunov function. Hint: x − sin x > 0 for x > 0, and x − sin x < 0 for x < 0. Consider these cases with y positive but y so small that y2 can be ignored compared to y. c.Using the energy function V(x,y)=12y2+(1−cos x)Vx,y=12y2+1−cos x mentioned in Problem 5b, show that the origin is a stable critical point. Since there is damping in the system, we can expect that the origin is asymptotically stable. However, it is not possible to draw this conclusion using this Liapunov function. d.To show asymptotic stability, it is necessary to construct a better Liapunov…20.Consider two interconnected tanks similar to those in Figure 7.1.6. Initially, Tank 1 contains 60 gal of water and Q01Q10 oz of salt, and Tank 2 contains 100 gal of water and Q02Q20 oz of salt. Water containing q1 oz/gal of salt flows into Tank 1 at a rate of 3 gal/min. The mixture in Tank 1 flows out at a rate of 4 gal/min, of which half flows into Tank 2, while the remainder leaves the system. Water containing q2 oz/gal of salt also flows into Tank 2 from the outside at the rate of 1 gal/min. The mixture in Tank 2 leaves it at a rate of 3 gal/min, of which some flows back into Tank 1 at a rate of 1 gal/min, while the rest leaves the system. a.Draw a diagram that depicts the flow process described above. Let Q1(t) and Q2(t), respectively, be the amount of salt in each tank at time t. Write down differential equations and initial conditions for Q1 and Q2 that model the flow process. b.Find the equilibrium values QE1Q1E and QE2Q2E in terms of the concentrations q1 and q2. c.Is it…