The system of differential equations 1 dx x dt 1 dy = 1- x - y y dt y 2 2' model the interaction of two populations x and y. (a) What kind of interaction do these equations describe? ? (Symbiosis takes place when the interaction of two species benefits both. An example is the pollination of plants by insects.) (b) Now suppose that we start with the initial populations (x(0), y(0)) = (1, 1.5). What happens to the populations in the long run? (For each, enter infinity or a numerical value.) x goes to y goes to

Solve the system of differential equations. 6 -2 - X' = X 2 10 x₁(0) = 18, x₂(0) - 26 x₁ (t) = x₂(t) = 8 =

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) =