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- 8.A spring–mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of (3 cos(3t) − 2 sin(3t)) N, determine the steady-state response. Express your answer in the form R cos(ωt − δ).1. Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. a) ?(? − 4)? ′′ + 3?? ′ + 4? = 2, ?(6) = 0, ? ′(6) = −1 b) (? + 1)? ′′ + ?? ′ + ? = sec ?, ?(0) = 2, ? ′(0) = −1 c) (? − 4)? ′′ + 3?? ′ + ln(?) ? = sin ?, ?(1) = −2, ? ′(1) = −1Max 1x1 + 1x2 s.t. 5x1 + 6x2 ≤ 41 1x1 + 5x2 ≤ 15 2x1 + 1x2 ≤ 15 x1, x2 ≥ 0 and integer Solve the LP Relaxation of this problem. at (x1, x2) =
- Two very large tanks A and B are partially filled with 100 gallons of brine each. Initially, 100 pounds of salt are dissolved in the solution in tank A and 50 pounds of salt are dissolved in the solution in tank B. The system is closed, since the well-mixed liquid is pumped only between the tanks as shown in the figure. 1. Use the information in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time "t" in tanks A and B, respectively. 2. Find a relationship between the variables x1(t) and x2(t) that holds at time «t». 3. Explain why this relationship makes intuitive sense. 4. Use this relationship to help find the amount of salt in tank B at t = 30 min.Consider the differential equationu'' + 6u' + 9u = 0 : It has solutions u = e^-3t and u = t e^-3t. A.1. Compute the Wronskian Wr[e^-t; t e^-3t](t). How can we know that Wr[e^-3t; t e^-3t](t) is proportional to e^6t before we compute it? (Hint: Abel.) A.2. Why are e^-3t and t e^-3t linearly independent functions? A.3. Find the natural fundamental set of solutions to the equation associated with t = 0.The indicated function y1(x) is a solution of the given differential equation. x2y'' + 2xy' − 6y = 0; y1 = x2(a) Use reduction of order or formula (5) in Section 4.2 to find a second solution y2(x).(b) Use the Wronskian to verify that the two functions y1(x) and y2(x)are linearly independent.
- A fish population in a lake satisfies dB/dt=cB(K-B) Fish are then added at A fish per hour. Find the DE for B with respect to t. Is there a new limiting population? If so, what is the solution?Find y''of 4cos(x/2)+e^-2xA continuous periodic signal x (t) of real value y has a fundamental period of T = 8. The coefficients of the nonzero Fourier series for x (t) are a1 = a-1 = 2, a3 = a * -3 = 4j Express x (t) in the form
- Question is in the image (question 3) FINDING THE GENERAL SOLUTION OF NONHOMOGENEOUS 2ND ODEsWhich of the following is TRUE for the rules of differential equations? (a) The integral of the ODE on an interval [a,b] must be differentiable for which its derivatives exist and satisfy the equation. (b) The constant C of the solution of the ODE must be specifically deter- mined to obtain the soluton generally. (c) The potential function of the ODE must be specified such that it must be given to determine the solutions of the general solutions. (d) The above statements are true if and only if the given equation is an ordinary differential equations.4. Find a solution of the Bessel's equation of order zero.