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1. (1 pt) Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations. y 0 1 = 5 2 y1 − 3 2 y2 y 0 2 = − 3 2 y1 + 5 2 y2 • A. y1 = e x y2 = e x • B. y1 = sin(x) +cos(x) y2 = cos(x)−sin(x) • C. y1 = e −x y2 = e −x • D. y1 = 2e −2x y2 = 3e −2x • E. y1 = e 4x y2 = −e 4x • F. y1 = cos(x) y2 = −sin(x) • G. y1 = sin(x) y2 = cos(x) As you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions. 2. (1 pt) Write the given second order equation as its equivalent system of first order equations. u 00 +6u 0 +3u = 0 Use v to represent the ”velocity function”, i.e. v = u 0 (t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u 0 = v 0 = Now write the system using matrices: d dt u v = u v . 3. (1 pt) Write the given second order equation as its equivalent system of first order equations. u 00+5.5u 0−7.5u = −6 sin(3t), u(1) = −7.5, u 0 (1) = −6 Use v to represent the ”velocity function”, i.e. v = u 0 (t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u 0 = v 0 = Now write the system using matrices: d dt u v = u v + and the initial value for the vector valued function is: u(1) v(1)  = . 4. (1 pt) Write the given second order equation as its equivalent system of first order equations. t 2 u 00 −4tu0 + (t 2 +6)u = 2.5 sin(3t) Use v to represent the ”velocity function”, i.e. v = u 0 (t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u 0 = v 0 = Now write the system using matrices: d dt u v = u v + . 5. (1 pt) Consider two interconnected tanks as shown in the figure above. Tank 1 initial contains 60 L (liters) of water and 370 g of salt, while tank 2 initially contains 100 L of water and 465 g of salt. Water containing 15 g/L of salt is poured into tank1 at a rate of 1.5 L/min while the mixture flowing into tank 2 contains a salt concentration of 45 g/L of salt and is flowing at the rate of 2 L/min. The two connecting tubes have a flow rate of 5.5 L/min from tank 1 to tank 2; and of 4 L/min from tank 2 back to tank 1. Tank 2 is drained at the rate of 3.5 L/min. You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving any tank along any of the tubes has the same concentration of salt as the tank as a whole. (This is not completely realistic, but as in real physics, we are going to work with the approximate, rather than exact description. The ’real’ equations of physics are often too complicated to even write down precisely, much less solve.) How does the water in each tank change over time? Let p(t) and q(t) be the amount of salt in g at time t in tanks 1 and 2 respectively. Write differential equations for p and q. (As usual, use the symbols p and q rather than p(t) and q(t).) p 0 = 1 q 0 = Give the initial values: p(0) q(0) = . 6. (1 pt) Consider the system of differential equations dx dt = −4y dy dt = −4x. Convert this system to a second order differential equation in y by differentiating the second equation with respect to t and substituting for x from the first equation. Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0) = 3 and y(0) = 2, what are x and y? x(t) = y(t) = 7. (1 pt) Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations, which correspond to the slope field shown below. dw dt = w−wr, dr dt = −r +wr. Assume w = 1 and r = 3 when t = 0. Does the number of worms increase, decrease, or stay the same at first? ? Does the number of robins increase, decrease, or stay the same at first? ? What happens in the long run? ? 8. (1 pt) Consider a conflict between two armies of x and y soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and t represents time since the start of the battle, then x and y obey the differential equations dx dt = −ay, dy dt = −bx, where a and b are positive constants. Suppose that a = 0.05 and b = 0.02, and that the armies start with x(0) = 49 and y(0) = 18 thousand soldiers. (Use units of thousands of soldiers for both x and y.) (a) Rewrite the system of equations as an equation for y as a function of x: dy dx = (b) Solve the differential equation you obtained in (a) to show that the equation of the phase trajectory is 0.05y 2 −0.02x 2 = C, for some constant C . This equation is called Lanchester’s square law . Given the initial conditions x(0) = 49 and y(0) = 18, what is C? C = Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 1. (1 pt) Compute the following product.  5 −4 1 −3 −2 −1 2 5 2 2. (1 pt) Compute the following product. −3 4 1 2 1 −1 3 −3 2 = 3. (1 pt) Compute the following product. −3 4 1 1 −1 3 2 3 −1 −1 = 4. (1 pt) If A = −2 0 1 3 3 −1 −4 0 −3 −2 2 −3 1 −2 4 −1 −4 −1 Then AB = and BA = 5. (1 pt) Match the differential equations and their vector valued function solutions: It will be good practice to multiply at least one solution out fully, to make sure that you know how to do it, but you can get the other answers quickly by process of elimination and just multiply out one row element. 1. y 0 (t) = −86 218 −160 73 −49 80 111 −138 165  y(t) 2. y 0 (t) = −13 −2 3 −15 −18 5 −33 −18 7 y(t) 3. y 0 (t) = −97 33 −5 −140 84 35 −4 15 −8 y(t) A. y(t) = -1 0 -3 e −4t B. y(t) = -2 1 3 e 45t C. y(t) = -2 -4 4 e −21t 1 1. (1 pt) Solve the system dx dt = −20 20 −5 5 x with the initial value x(0) = -15 -3 . x(t) . 2. (1 pt) Consider the system of differential equations dx dt = −1.6x+0.5y, dy dt = 2.5x−3.6y. For this system, the smaller eigenvalue is and the larger eigenvalue is . Use the phase plotter pplane9.m in MATLAB to determine how the solution curves behave. • A. The solution curves race towards zero and then veer away towards infinity. (Saddle) • B. All of the solution curves run away from 0. (Unstable node) • C. All of the solution curves converge towards 0. (Stable node) • D. The solution curves converge to different points. The solution to the above differential equation with initial values x(0) = 5, y(0) = 6 is x(t) = , y(t) = . 3. (1 pt) Consider the systems of differential equations dx dt = 0.4x+0.5y, dy dt = 1.5x−0.6y. For this system, the smaller eigenvalue is and the larger eigenvalue is . Use the phase plotter pplane9.m in MATLAB to determine how the solution curves behave. • A. The solution curves race towards zero and then veer away towards infinity. (Saddle) • B. All of the solution curves converge towards 0. (Stable node) • C. The solution curves converge to different points. • D. All of the solution curves run away from 0. (Unstable node) The solution to the above differential equation with initial values x(0) = 3, y(0) = 3 is x(t) = , y(t) = . 4. (1 pt) Consider the systems of differential equations dx dt = 0.1x−0.8y, dy dt = −0.2x+0.7y. For this system, the smaller eigenvalue is and the larger eigenvalue is . Use the phase plotter pplane9.m to determine how the solution curves behave. • A. All of the solution curves run away from 0. (Unstable node) • B. All of the solution curves converge towards 0. (Stable node) • C. The solution curves race towards zero and then veer away towards infinity. (Saddle) • D. The solution curves converge to different points. The solution to the above differential equation with initial values x(0) = 8, y(0) = 2 is x(t) = , y(t) = . 1 1. (1 pt) Solve the system dx dt =  −3 3 −6 3 x with x(0) = 5 6  . Give your solution in real form. x1 = , x2 = . ? 1. Use the phase plotter pplane9.m in MATLAB to describe the trajectory. 2. (1 pt) Solve the system dx dt =  −4 −2 2 −4  x with x(0) = 4 2  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane9.m in MATLAB to answer the following question: ? 1. Describe the trajectory. 3. (1 pt) Solve the system dx dt =  −4 −2 10 4  x with x(0) = 4 4  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane9.m in MATLAB to answer the following question. ? 1. Describe the trajectory. 4. (1 pt) Solve the system dx dt =  3 −2 2 3  x with x(0) = 8 4  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane9.m in MATLAB to answer the following question. ? 1. Describe the trajectory. 1 1. (1 pt) Solve the system dx dt =  1 −4 4 −7  x with x(0) = 3 2  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane7.m in MATLAB to determine how the solution curves (trajectories) of the system x 0 = Ax behave. • A. All of the solution curves converge towards 0. (Stable node) • B. The solution curves race towards zero and then veer away towards infinity. (Saddle) • C. The solution curves converge to different points. • D. All of the solution curves run away from 0. (Unstable node) 2. (1 pt) Solve the system dx dt =  2 1.5 −1.5 −1  x with x(0) = 3 -2  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane7.m in MATLAB to determine how the solution curves (trajectories)of the system x 0 = Ax behave. • A. All of the solution curves run away from 0. (Unstable node) • B. All of the solution curves converge towards 0. (Stable node) • C. The solution curves race towards zero and then veer away towards infinity. (Saddle) • D. The solution curves converge to different points. 3. (1 pt) Solve the system dx dt =  −2.5 1.5 −1.5 0.5  x with x(0) = 3 -1  . Give your solution in real form. x1 = , x2 = . Use the phase plotter pplane7.m in MATLAB to determine how the solution curves of the system x 0 = Ax behave. • A. All of the solution curves converge towards 0. (Stable node) • B. All of the solution curves run away from 0. (Unstable node) • C. The solution curves converge to different points. • D. The solution curves race towards zero and then veer away towards infinity. (Saddle) 4. (1 pt) Solve the system dx dt =  3 9 −1 −3 x with x(0) = 2 4 . Give your solution in real form. x1 = , x2 =

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