You are given the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form: x ̇ = 2 x + y + 3 e t, y ̇ = 4 x − y . Write down the complementary function for the system of equations and then find a particular integral for the original inhomogeneous system.

Express the system of differential equations in matrix notation. - x" -6x+ty (sint)x = 0 y'"'+5y"-tx' + 3y' + e¯¹x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. x₁ = x, x₂ = x², x3 = y, x4 = y' OB. X₁ =x', x=x", x3 = y', x4 = y", x5 =y"" OC. x₁ = x, x2 =x', x3 = y, x4 = y', x5 =y" OD. x₁ = x, x₂ =x", x3 = y, x4 = y'"' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) x1 X1 x2 X3 ○ A. ха X5 X5 ہیں ○ B. کا 5 x1 x2 X3 ха

This is the first part of a two-part problem. Let P=[-: 1 5₁(t) = [(41) 5₂(t) = - sin(4t). a. Show that y₁ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product y(t) = = 0 [1] -4 Enter your answers in terms of the variable t. -4 sin(4t) -4 cos(4t)] ÿ₁ (t) [181-18] b. Show that y₂ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product Enter your answers in terms of the variable t. 04] 32(t) = [-28]|2(t) 181-181

You are given the following inhomogeneous system of first-order differentialequations for x(t) and y(t) in matrix form: x ̇ = 2x + y + 3 et ,y ̇ = 4x − y  What is the long-term behaviour of this particular solution as t becomes large? Does the ratio y/x tend to a fixed number, and if so what number?