odinary differential equations

For the ODE 2 dy + 3xy = e-1.5x dx y(0) = 3, estimate y(4) with a step size of 2 using a. Heun's Runge-Kutta 2nd order method b. Midpoint Runge-Kutta 2nd order method c. Ralston's Runge-Kutta 2nd order methodD Question 30 50 pt Regression 1) Regression Analysis: Concession 1000 versus Attendee100 N-24 Model Summary S R-sq R-sqladj) R-sqlpred) 57.4275 0.05% 0.00% 0.00% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 231.7 41.6 5.57 0.000 Attendee100 0.15 1.49 0.10 0.921 1.00 Durbin-Watson Statistic Durbin-Watson Statistic = 0.178614 Regression 2) Regression Analysis: Concession 1000 versus Attendee100, Time N-25 Model Summary R-sqladil R-sqlpred e e w 9Consider a birth and death process X(t), t 2 0, such as the branching process, that has state space {0, 1, 2, ...} and birth and death rates of the form Ax = x1 and Hx = XH, x 2 0, where 1 and u are nonnegative constants. Set my(t) = E.(X(t)) = > yP x,(t). )=0 (a) Write the forward equation for the process. (b) Use the forward equation to show that my(t) = (2 - u)m.(t). (c) Conclude that my(t) = xe(2-4)tConsider a birth and death process with birth rates ); = (i + 1)2, i 2 0, and death rates u; = iu, i 2 0. (a) Determine the expected time to go from state 0 to state 4. (b) Determine the expected time to go from state 2 to state 5. (c) Determine the variances in parts (a) and (b).Exercises 4.1. In a birth and death process with birth parameters A,, = A for n = 0, 1, . . . and death parameters /,, = un for n = 0, 1, . . ., we have Po, (t) = (Ap )'exp j! where p= -[1 - e-"g]. Verify that these transition probabilities satisfy the forward equations (4.4), with i = 0. Pho(t) = - doPi.o ( t) + M, Pil (1), (4.4) P'., (1) = >;-P.j-(1) - (1; + 1, ) P., (1 ) + Me;+ P.j + (1), jz1