First order difference equation Consider the simple model |y(t)+ay(t-1)=bu(t-1) ÔN Σy²(t-1) -Σy(t-Du(t-1)] -Σy(t-1)u(t-1) Σu²(t-1) HIE -Σy(t)y(t-1)] Σy(t)u(t-1) Exercise: Suppose for t=1 to 6 the value of u and y are: u=[2 3 2 5 3 7 y=[-23 48 -93 188 -371 745 Derive N
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![First order difference equation
Consider the simple model
y(t)+ay(t-1)=bu(t-1)
0
Σy²(t-1)
-Σy(t-1)u(t-1) Σu'(1-1)
-Ey(t-Du(t-1)-Ey(t)y(t-1)
Σy(t)u(t-1)
-TE
Exercise: Suppose for t=1 to 6 the value of u and y are:
u=[2 3 2 5 3 7 y=[-23 48 -93 188 -371 745]
Derive Ov](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F321ef1f9-4de3-43d5-9e6a-51e883aeda71%2F7f527041-fe23-4705-9313-f656d6ada83e%2F6iv7on_processed.jpeg&w=3840&q=75)
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- A mixing tank A 500-liter (L) tank is filled with pure water. Attime t = 0, a salt solution begins flowing into the tank at a rate of5 L/min. At the same time, the (fully mixed) solution flows outof the tank at a rate of 5.5 L/min. The mass of salt in grams in thetank at any time t Ú 0 is given byM(t) = 250(1000 - t)(1 - 10-30(1000 - t)10)and the volume of solution in the tank is given byV(t) = 500 - 0.5t.a. Graph the mass function and verify that M(0) = 0.b. Graph the volume function and verify that the tank is emptywhen t = 1000 min.c. The concentration of the salt solution in the tank (in g/L) isgiven by C(t) = M(t)>V(t). Graph the concentration functionand comment on its properties. Specifically, what are C(0)and lim t-S1000- C(t)?d. Find the rate of change of the mass M′(t), for 0 <, t < 1000.e. Find the rate of change of the concentration C′(t), for0 < t < 1000.f. For what times is the concentration of the solution increasing?Decreasing?Differential equations. Autonomous systems and stability.Suppose that the solution (x1(t),y1(t)) is defined for all t and that its trajectory has an apparent self-intersection x1(a) = x1(a+p) = x0 y1(a) = y1(a+p) = y0 With p>0. with the solution x2(t) = x1(t+p) y2(t) = y1(t+p) And applying the uniqueness theorem, prove: x1(t+p) = x1(t) y1(t+p) = y1(t) for all t. That is, the solution is closed and has a period p. Consequently the solution to an autonomous system is periodic with a closed path, or never passes through the same point twice. Please be as clear as possible, and legible. Show and explain all the steps in detail and use definitions if necessary. Thank you very much.12 help use variation of parameters to finda general solution to the differntial equation given the function y1 and y2 are linearly independent solutions to the corresponding homogenous equation for t>0
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- two linearly independent solutions y1 and y2are given to the associated homogeneous equation of the variablecoefficientnonhomogeneous equation. Use the method of variation ofparameters to find a particular solution to the nonhomogeneous equation.Assume x > 0 in each exercise. x2y'' + 2xy' - 2y = x2, y1 = x -2, y2 = xDifferential equations Your employer automatically puts 10 percent of your salary into a 401(k) retirement account each year. The account earns 4% interest. Suppose you just got the job, your starting salary is $55000, and you expect to receive a 5% raise each year. Additionally, you decide to deposit a fixed amount of $4000 each year to play it conservatively.For simplicity, assume that interest earned, your raises, and your annual deposit all occur continuously.A) Let R(t) denote the dollar value of your retirement account in years. Write the differential equation modeling your account:dR/dt= B) Solve the differential equation to find R(t). Hint: make use of the Extended Linearity Principle.R(t)= C) Find the value of your retirement account after 35 years. Round answer to 2 decimal places.Value = $
