I need answers for questions 7-9

You notice that the flow of the water alternates between two flows that we'll call flow 1 and flow 2. The x and y coordinates are as shown above. The velocity v of the water is given by v = (u, v) = (-2,y) flow 1 lasting for ln(2) units of time and v=(u, v) = (-4, x) lasting for a units of time, where a is a constant. flow 2 Now you want to predict where you would end up after many alternating cycles. If your x-position at time t is the function X(t) and your y-position at time t is the function Y(t), then your position as a function of time satisfies the pair of ODES dx dY and dt dt Note that the value of u and v at a given time depends on your position at that time. =U =U 4. Write down the corresponding ODEs for flow 1 and solve for X(t) and Y(t) given an initial location X = X, and Y = Y, at time t = 0. 5. Let X1 = (X1,Y1) = (X(ln(2)), Y(ln(2))) be your position vector at time t = ln(2). Show that it can be written in terms of a matrix transformation X1 = AX where Xo = (Xo, Yo), and find the matrix A. 6. Now write down the ODEs for flow 2 and solve for X(t) and Y(t) given an initial location of X = X1 and Y = Y at time t = ln(2). You may find it useful to substitute X(t) = r(t) cos[0(t)] and Y(t) r(t) sin[0(t)] to find ODEs for r(t) and e(t), solve these and then return to X(t) and Y(t) instead of working with the ODEs for X(t) and Y(t) directly. 7. Let X2 = (X2, Y2) = (X(ln(2) + a), Y (ln(2) + a)) be the position vector at time t = ln(2) + a. Show that it can be written as a matrix transformation X2 = BX1 and find the matrix B. 8. Hence find the matrix C that maps the initial location Xo to the position X2. How would you find the ion vector X2n after the nth occurrence of flow 2 in terms of C? You notice a knob that allows you to switch how long flow 2 lasts between a = a/ 7/2 and a = a units of time. The monster reaches the platform and you jump into the water just as flow 1 starts. Your group jumps off one side of the platform and the monster off the other side so that your initial point is (10,1) and the monster's is (10, -1). The monster's arms can reach you at this distance, but before it can, the flow has begun to carry you and the monster. While the water moves, you and the monster are unable to swim. 9. For each value of a, how far away are you from the monster after an even number n=2m occurrences of the two flows? Assuming you want to escape from the monster, how should you have adjusted a before jumping? sol: Flow 1: voy dx dz and dy dt 17. - fat and sat Inx -t & Inxo lny ++ in yo - Xo et 15) At t. In 2, x=x, and - Ina Xo (21) = xo 2 and Y = Lena) Yo 240 X012 0 2 Yo 6 2 Transformation Matrin x = A xo :) :) (6) flow 2: uong von dx dt and dt Let y(t) r(t) cos o (6) Y(t) .YC) sin (D) : dx y coso ursing dl dy y sin o tycose JE [ada ] Los o -r sing or sino - 0 and & sino trcoso gercoso and equation multiply with sino equation multiply with cos o that both equation are after Subtract i sino coso - r sin o - r sino garsinko Sino coso y costo -y cesto + 0 + rccos?o - sin o) e r (costosinha) do dt 0 : ++ A 1: A is constant] from equation Los orsino -vsino .: cosoo [*] T: 8 [B is constant] X(t) B (09 (t+A), 4(4): 8 sin (4+A) now at t = ln 2, xax, and yoy, . X, B cos (In 2*A) , Y, B sin (Ina +A) now x2 + 7,2 B B = x," + Y, 2 Also tan (In 2 +A) an X(t) f***?. cos (t + tari" ) - in 2) and y(t)=14,2+y, sin (t+ tan.) - in2) According to your request 4, 5, 6 are, solved. You notice that the flow of the water alternates between two flows that we'll call flow 1 and flow 2. The x and y coordinates are as shown above. The velocity v of the water is given by v = (u, v) = (-2,y) flow 1 lasting for ln(2) units of time and v=(u, v) = (-4, x) lasting for a units of time, where a is a constant. flow 2 Now you want to predict where you would end up after many alternating cycles. If your x-position at time t is the function X(t) and your y-position at time t is the function Y(t), then your position as a function of time satisfies the pair of ODES dx dY and dt dt Note that the value of u and v at a given time depends on your position at that time. =U =U 4. Write down the corresponding ODEs for flow 1 and solve for X(t) and Y(t) given an initial location X = X, and Y = Y, at time t = 0. 5. Let X1 = (X1,Y1) = (X(ln(2)), Y(ln(2))) be your position vector at time t = ln(2). Show that it can be written in terms of a matrix transformation X1 = AX where Xo = (Xo, Yo), and find the matrix A. 6. Now write down the ODEs for flow 2 and solve for X(t) and Y(t) given an initial location of X = X1 and Y = Y at time t = ln(2). You may find it useful to substitute X(t) = r(t) cos[0(t)] and Y(t) r(t) sin[0(t)] to find ODEs for r(t) and e(t), solve these and then return to X(t) and Y(t) instead of working with the ODEs for X(t) and Y(t) directly. 7. Let X2 = (X2, Y2) = (X(ln(2) + a), Y (ln(2) + a)) be the position vector at time t = ln(2) + a. Show that it can be written as a matrix transformation X2 = BX1 and find the matrix B. 8. Hence find the matrix C that maps the initial location Xo to the position X2. How would you find the ion vector X2n after the nth occurrence of flow 2 in terms of C? You notice a knob that allows you to switch how long flow 2 lasts between a = a/ 7/2 and a = a units of time. The monster reaches the platform and you jump into the water just as flow 1 starts. Your group jumps off one side of the platform and the monster off the other side so that your initial point is (10,1) and the monster's is (10, -1). The monster's arms can reach you at this distance, but before it can, the flow has begun to carry you and the monster. While the water moves, you and the monster are unable to swim. 9. For each value of a, how far away are you from the monster after an even number n=2m occurrences of the two flows? Assuming you want to escape from the monster, how should you have adjusted a before jumping? sol: Flow 1: voy dx dz and dy dt 17. - fat and sat Inx -t & Inxo lny ++ in yo - Xo et 15) At t. In 2, x=x, and - Ina Xo (21) = xo 2 and Y = Lena) Yo 240 X012 0 2 Yo 6 2 Transformation Matrin x = A xo :) :) (6) flow 2: uong von dx dt and dt Let y(t) r(t) cos o (6) Y(t) .YC) sin (D) : dx y coso ursing dl dy y sin o tycose JE [ada ] Los o -r sing or sino - 0 and & sino trcoso gercoso and equation multiply with sino equation multiply with cos o that both equation are after Subtract i sino coso - r sin o - r sino garsinko Sino coso y costo -y cesto + 0 + rccos?o - sin o) e r (costosinha) do dt 0 : ++ A 1: A is constant] from equation Los orsino -vsino .: cosoo [*] T: 8 [B is constant] X(t) B (09 (t+A), 4(4): 8 sin (4+A) now at t = ln 2, xax, and yoy, . X, B cos (In 2*A) , Y, B sin (Ina +A) now x2 + 7,2 B B = x," + Y, 2 Also tan (In 2 +A) an X(t) f***?. cos (t + tari" ) - in 2) and y(t)=14,2+y, sin (t+ tan.) - in2) According to your request 4, 5, 6 are, solved