Calculus IV - HW 2 MA 214. Due 6/29

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1 Calculus IV - HW 2 MA 214 Due 6/29 Section (Problems 3 and 5 from B&D) The following problems involve differential equations of the form dy = f(y). For each, sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. (a) dy = y(y 1)(y 2); y 0 0 (b) dy = e y 1; < y 0 < 2. (Problems 9 and 13 from B&D) The following problems involve differential equations of the form dy = f(y). For each, sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable, unstable, or semistable. Draw the phase line, and sketch several graphs os solutions ain the ty-plane. (a) dy = y2 (y 2 1); (b) dy = y2 (1 y) 2 ; < y 0 < < y 0 < 3. (Problems 16 from B&D) Another equation that has been used to model population growth is the Gompertz equation dy where r and k are positive constants. = ry ln(k/y) (a) Sketch the graph of f(y) versus y, find the critical points, and determine whether each is asymptotically stable or unstable. (b) For 0 y K, determine where the graph of y versus t is concave up and where it is concave down. 1

2 (c) For each y in 0 < y K, show that dy as given by the Gompertz equation is never less than dy dy as given the the logistic equation = r( ) 1 y K y. 4. (Problem 17a from B&D) Solve the Gompertz equation dy subject to the initial condition y(0) = y 0. Hint: You may want to let u = ln(y/k). = ry ln(k/y) 5. Optional Hard Problem (Problems 18 from B&D) A pond forms as water collects in a conical depression of radius a and depth h. Suppose that water flows in at a constant rate k and is lost through evaporation at a rate proportional to the surface area. (a) Show that the volume V (t) of water in the pond at time t satisfies the differential equation dv 3a = k απ( πh ) V 3 where α is the coefficient of evaporation. (b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable? (c) Find a condition that must be satisfied if the pond is not to overflow. 6. Suppose we are given an autonomous differential equation dy graph of f(y) vs y. = f(y) with the following 1.jpg (a) Find all the critical points, draw the phase lines of the graph, and determine whether each critical point cooresponds to a stable, unstable, or semistable equilibrium solution. Page 2

3 (b) Between each pair of critical points, estimate the y value at which solutions whose initial values are in that interval switch concavity. (c) Use parts (a) and (b) to sketch solutions to the differential equation in the ty plane given different initial conditions y(0) = y 0. Include several solutions with initial conditions y(0) = y 0 between each pair or equilibrium. solutions. (d) Recall that a threshold for a population is value above which the population grows but below which the population becomes extinct. Use part (c) to determine the threshold and carrying capacity for the population described by this differential equation. (e) Based off your investigations and the the discussion on page 86 and 87 of B&D, give a function for f(y). Section (Problems 1,5,6,8 from B&D) For the following problems, determine whether or not the equations are exact. You do not need to solve the equations. (a) (2x + 3) + (2y 2)y = 0 (b) dy = ax+by dx bx+cy (c) dy = ax by dx bx cy (d) ( e x sin(y) + 3y ) dx ( 3x e x sin(y) ) dy = 0 8. Show that the following equations are exact and use this to find an implicit solution for y. (a) (2xy + 5y 3 ) + (x xy )y = 0 (b) ( sin(x)y xy 2 + e x ) + (2y cos(x) + 30x 2 y)y = 0 9. Put the equation y = e 2x + y 1 into the standard form for exact equations and show that the equation is not exact. Determine which of the following integration factors make the equation exact. You don t need to solve the equation. (a) µ(t) = e y (b) µ(t) = e x (c) µ(t) = y 10. (Problems 25 and 27 from B&D) For each of the following, find an integrating factor and solve the given equation. Page 3

4 (a) ( 3x 2 y + 2xy + y 3 )dx + (x 2 + y 2 )dy = 0 (b) dx + ( x y sin(y)) dy = Consider the differential equation with initial condition y(0) = y 0. (3x 2 y + 10e x + 4y 2 ) + (x 3 + cos(y) + 8xy)y = 0 (a) Show that the equation is exact. (b) Solve the equation in the following ways: i. Integrate M(x, y) with respect to x and find the function h(y) such that ψ = M(x, y)dx + h(y). ii. Integrate N(x, y) with respect to y and find the function g(x) such that ψ = N(x, y)dy + g(x). 12. Consider the differential equation y dx+(2xy e 2y ) dy = 0. Show that it is not exact, find an integrating factor, and use this to find an implicit solution to the equation. Section (Problem 1 from B&D) Consider the initial value problem y = 3 + t y y(0) = 1 (a) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h = 0.1. (b) Repeat part (a) with h = Compare the results with those found in (a). (c) Find the solution y = φ(t) of the given problem and evaluate φ(t) at t = 0.1, 0.2, 0.3, and 0.4. Compare these values with the results of (a) and (b). 14. (Problem 12 from B&D) Consider the initial value problem y = y(3 ty) y(0) = 0.5 Use Euler s method to find approximate values of the solution at t = 0.5, 1.0, 1.5, 2.0 (a) with h = 0.5 (b) with h = 0.25 Page 4

5 15. Optional Hard Problem Solve the differential equation ( 3xy + y 2 ) + ( x 2 + xy ) y = 0 using the integrating factor µ(x, y) = [xy(2x + y)] 1. Verify that the solution is the same as that obtained using a different integrating factor in the lecture notes (example 4 from B&D). 16. (Problem 2 from B&D) Consider the initial value problem y = 2y 1 y(0) = 1 (a) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h = 0.1. (b) Repeat part (a) with h = Compare the results with those found in (a). (c) Find the solution y = φ(t) of the given problem and evaluate φ(t) at t = 0.1, 0.2, 0.3, and 0.4. Compare these values with the results of (a) and (b). 17. Consider the autonomous equation dy = ry where r > 0. If we apply Euler s method with step size h, we notice that the equation y n = h f(t n 1, y n 1 ) + y n 1 becomes y n = h f(y n 1 ) + y n 1. Suppose we are given the initial condition y(t o ) = y 0. (a) Show that y i = y i 1 (1 rh). (b) Use (a) to show y n = y 0 (1 rh) n. (c) Optional Hard Problem We say a family of solution converges if as t, the solutions approach the same value regardless of initial condition. i. For what values of h do the approximations given by Euler s method converge? ii. Solve the differential equation explicitly. You should note that all its solutions converge. iii. Why is this not true for the approximations given by Euler s method if h is too large? (It may be helpful to consider the slope fields of the equation.) 18. (Problems 1,5,6,8 from B&D) For the following problems, determine whether or not the equations are exact. You do not need to solve the equations. (a) (2x + 3) + (2y 2)y = 0 (b) dy = ax+by dx bx+cy (c) dy = ax by dx bx cy (d) ( e x sin(y) + 3y ) dx ( 3x e x sin(y) ) dy = Show that the following equations are exact and use this to find an implicit solution for y. Page 5

6 (a) (2xy + 5y 3 ) + (x xy )y = 0 (b) ( sin(x)y xy 2 + e x ) + (2y cos(x) + 30x 2 y)y = Put the equation y = e 2x + y 1 into the standard form for exact equations and show that the equation is not exact. Determine which of the following integration factors make the equation exact. You don t need to solve the equation. (a) µ(t) = e y (b) µ(t) = e x (c) µ(t) = y 21. (Problems 25 and 27 from B&D) For each of the following, find an integrating factor and solve the given equation. (a) ( 3x 2 y + 2xy + y 3 )dx + (x 2 + y 2 )dy = 0 (b) dx + ( x y sin(y)) dy = 0 Section Find the general solution to the differential equation y + y 30y = Find the solution to the following initial value problems: (a) y + 4y + 3y = 0; y(0) = 2, y (0) = 0 (b) y 6y + 8y = 0; y(0) = 3, y (0) = Find a second order linear equation with constant coefficients whose general solution is: (a) C 1 e 2t + C 2 e 5t (b) C 1 e 3t + C 2 e t Page 6