Differential Equations a) Transform the equation into a system of first order equations: y"+5y'+2y=3sin(t) b) Transform the system with the initial conditions into a single second order equation: x1'= 3x1-2x2, x1(0)=3 x2'=2x1-2x2, x2(0)=1/2

where A system of system of first-order linear differential equations has the form y' = Ay y' = , y = Y1 Y2 , A = a11 a21 This gives us the system B [an an2 ann] Solve the system of linear differential equations by following the steps listed. y₁ = y₁ 432 y₂ = -2y1 + 8y2 a12 022 ain a2n ⠀ (a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A should be in this matrix equation. (b) We would like to have the equations so that y is a function of only y₁ and ₂2 is a function of only y2. To do this, we want to turn A into a diagonal matrix. This can be done by taking the diagonalization of A. Diagonalize the matrix A you created in part a. (c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and y to be Pw' = y Pw=y Pw = APw Since P is invertible, we can multiply P-1 to the left on both sides of the equations to get w = P-¹APW = Dw Write out what the system of linear differential equations looks like now. (d) Let k be any scalar.…

Solve the system of differential equations Jx¹= x'= - 4x + 21y y=0x + 3y x(0) = 6, y(0) = 2 x(t) = y(t) =

A system is described by the differential equation –2y" (t) – 5y' (t) + 5y(t) = ys(t), Find the transfer function associated with this system H(s). Write the solution as a single fraction in s H(s) = help (formulas)

Solve the system of differential equations Jx'= -23x + 72y y'= -6x + 19y x(0) = -6, y(0) = -1 x(t) = y(t) =