Consider the system of higher order differential equations y″ = t¯¹y' + 7y – tz + (sint)z' + e5t, z" = y — 3z'. Rewrite the given system of two second order differential equations as a system of four first order linear differential equations of the form ÿ' = P(t)ÿ+ g(t). Use the following change of variables y' [Y] Y1 Y2 Y3 LY4_ help (formulas) help (matrices) Book: Section 3.3 of Notes on Diffy Qs [y1(t)] [ y(t)] Y2(t) y' (t) ÿ(t) = = Y3(t) z(t) Y₁(t)] [z'(t)]

Convert the differential equation u''+5u'−3u=e−4tu′′+5u′-3u=e-4t into a system of first order equations by letting x=u, y=u'x=u, y=u′x'x′ =   y'y′ =

Convert the differential equation u'' + 3u' + 2u = e x = u, y = u' 4t into a system of first order equations by letting x' = y' =

Convert the differential equation u"+2u'+5u = et into a system of first order equations by letting x = u, y=u' x' = y'

3t Convert the differential equation u"+u'-u = e³t into a system of first order equations by letting x = u, y = u' x' = y' =

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Solve the system [: 16 -36 dx dt 6. -14 with the initial value 13 x(0) 5 -5e^(4t)+13e^(-3t) x(t) = -5e^(4)+10e^(-3t)

The Kermack-McKendrick equations are first order differential equations describing an infectious disease outbreak. Using S to denote the number of susceptible members of the population and I to denote the number of infected members, the equations are S' = - BSI I' = BSI – }I, where 3 and μ are positive constants representing the transmission rate and rate of recovery. Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane.

Find the general solution to the differential system: ས་ 1) dt (G) (19) (G)

Match the differential equation with its direction field. y' = 9(x + y) - 1 | y y y y 0.3 15 0:2 / 0:2+ 0.1+ 0:1- /// / // -Q.3) -Q.21 -Q.1\ \ 01- -0.2 /0,3 / -0.3- -0.2- -0.1- 0.1 1- -0.2 - -0.3 //// //// -0:1- مر م م مر محر //// ///// / //// //// // 0 for 0 < x < r/9, 0 < y < n/9. y' = 9(x + y) – 1 = 0 on the line y = -x + 1/9, and y' = -1 on the line y = -x. y' = 9(x + y) – 1 = 0 on the lines x = 0 and y = 9. The slopes at each point are independent of x, so the slopes are the same along each line parallel to the x-axis. Note that for y = 9, y' = 0. The slopes at each point are independent of y, so the slopes are the same along each line parallel to the y-axis. Note that for y = 9, y' = 0. %3D