Additional Homework Problems

Transcription

1 Additional Homework Problems These problems supplement the ones assigned from the text. Use complete sentences whenever appropriate. Use mathematical terms appropriately. 1. What is the order of a differential equation? 2. What does it mean to solve an ODE? 3. What is an initial value problem? What does it mean to solve an initial value problem? 4. Is it possible for a single function to solve many different ODEs? Explain. 5. What is a slope field (direction field)? Explain precisely how it is constructed. 6. Suppose you are given a function f(y) and you know that for any (t 0,y 0 ), the ODE y (t) = f(y(t)) has a unique solution y(t) in a neighborhood of (t 0,y 0 ) and satisfying the initial condition y(t 0 ) = y 0. Now, your friend sketches a graph of two solutions to the ODE, starting from two different points (t 0,a) and (t 0,b), a b. At some point in the plane, the graphs that your friend has drawn intersect each other. You protest, That cannot be right! Your friend responds defensively, Of course it can be, the curves start from different points! Explain who is right and why. 7. Consider the following initial value problem: y (t) = f(t,y), y(t 0 ) = b, y (t 0 ) = c. (0.1) where f is continuously differentiable, and b and c are some numbers. Explain why this problem may not have a solution. 8. What is the fundamental theorem of calculus and how is this used in solving separable equations? 9. What is an integrating factor? 10. What general types of first-order equations have we solved explicitly or implicitly? Give examples of each type. 11. Explain carefully why every separable equation is an exact equation. 12. Give an example of a first order equation which is exact but not separable. 13. Give an example of a first order equation which is not exact. 14. What is the difference between a linear first order ODE and a nonlinear first order ODE? Give examples. Author: James Nolen. Please report typos/errors to: nolen@math.duke.edu 1

2 15. Give an example of a function f(y) such that the solution to the initial value problem has all of the following properties: 16. Consider the initial value problem y (t) = f(y), t > 0; y(0) = a, (i) lim y(t) = 5, if a ( 1,5) t (ii) lim y(t) =, if a < 1 t (iii) y(t) is decreasing for all t > 0, if a > 5. y (t) = f(t,y(t)), y(t 0 ) = y 0. Find an example of a function f and a pair (t 0,y 0 ) such that the problem does not have a unique solution. In each example, explain where one or more of the hypotheses of the existence/uniqueness theorem break down. Hint: you might consider functions of the form f(y) = y p for certain powers p. 17. Give an example for which f(t,y) and f y (t,y) are continuous everwhere in the (t,y) plane, but a solution y(t) does not exist for all t. Why does this not contradict the theorem about existence and uniqueness? 18. Consider the initial value problem y (t) = y 2, y(0) = 2. (i) Suppose you apply Euler s method with step size h > 0 to approximate a solution. Let ỹ(t) be the approximate solution. Compute y 2 = ỹ(2h), the value of the approximate solution at time 2h (after 2 steps of the method). (ii) Will ỹ(h) (the approximation after 1 step) be above or below y(h) (the true solution)? (iii) Is it possible that ỹ(h) < 0? Is y(h) < 0? 19. Consider the initial value problem y +ty = t 2 y 2, y(0) = y 0. (i) Which of the following terms apply to this equation: linear, nonlinear, separable, exact? (ii) Suppose there is a solution y(t). What ODE does the function u(t) = (y(t)) 1 satisfy? (iii) Sove the ODE from (ii), and use this to write an integral expression for y(t). 20. Find a continuously differentiable function f(y) such that the solution of the initial value problem dy dt = f(y), y(0) = y 0 satisfies { 4, if lim y(t) = y0 > 3 t 0, if y 0 < 3. Be sure to explain why these limits hold for y(t), with this choice of f. 2

3 21. What is an autonomous equation? 22. What is an equilibrium solution? How do you find an equilibrium solution? 23. What is the difference between stable and unstable equilibrium solutions? Give an example of a function f(y) for which the equation y (t) = f(y(t)) has two stable equilibria and only one unstable equilibrium. 24. What is special about homogeneous equations having the form y (x) = f(y/x)? 25. How was our discussion of exact equations related to what you learned about gradient vector fields (conservative vector fields)? 26. What does it mean for a solution to be implicitly defined (rather than explicitly defined)? 27. Suppose you find an implicit solution y(t) to a first order ODE by finding a function H(y,t) such that H(y(t),t) = 0 for all t in the domain. Suppose your friend tries to solve the same ODE and comes up with a different function F(y,t) such that F(y(t),t) = 0 for all t in the domain. Could you both be right, or must one of you be wrong? 28. Suppose y(x) satisfies dy 3y 2x = dx 2y 3x Find an equation in x and y that defines y(x) implicitly. 29. Give an example of an equation for which Euler s method gives the exact solution (without error, for all step sizes t). 30. Give an example of an equation y (t) = f(y) and initial condition y(0) = y 0 for which Euler s method always gives values ȳ(t n ) that are greater than the true solution y(t n ), now matter how small the step size t. 31. What is the role of the step size h = t n+1 t n in Euler s method? Why is it important that the step size be sufficiently small? 32. For autonomous equations y (t) = f(y) why is it useful to know the intervals on which f > 0 or f < 0? 33. If the graph of the function y(x) is a level curve of the function ψ(x,y), what ODE must y solve? 34. Suppose that the graph of the function y(t) for t [ 2,2] is the upper half of a circle, centered at the point (0,3) and having radius 2. What ODE does y(t) satisfy? 35. Gather examples of each type of first order equation that we solved. Mix them up. Trade examples with a classmate and practice identifying which class of equations each example belongs to (e.g. linear, nonlinear, separable, nonseparable, exact, autonomous, homogeneous, none of the above). How would you solve each equation? 3

4 36. What does it mean for two functions y 1 (t) and y 2 (t) to be linearly independent? 37. What is the linear superposition principle? 38. Suppose that y 1 (t) and y 2 (t) both satisfy the linear 2nd order, homogeneous equation Ly = 0. Must any solution have the form y = c 1 y 1 +c 2 y 2? 39. Explain how eigenvalues of matrices arise naturally in the study of linear first order systems of ODEs. 40. Find three linearly-independent solutions to the linear, homogeneous third-order ODE y +2y y 2y = What is the reduction of order method and when is it useful? 42. Consider the equation d 2 y dt 2 2 ( ) dy 2 t dt + t 2 1 y = 0. The function y 1 (t) = te t satisfies the equation for t > 0. Find another solution y 2 (t) for t > 0 which is linearly independent of y 1. You may leave an integral in the expression for y 2 (t). 43. Find a particular solution to the equation 44. Consider the initial value problem y 3y 4y = 2sin(t). y (t) = cos(y (t)), y(0) = α, y (0) = β. Describe the qualitative behavior of y(t) as t +. The answer may depend on the constants α and β. You do not have to solve the equation or prove that a solution exists for all t (it does), but you must justify your answer. It may be useful to think about the function w(t) = y (t). 45. Solve the initial value problem What happens as t? y dy dt 1 y 3 = t2 y3, t 0, y(0) = Suppose we want to solve the initial value problem y +by +cy = f(t), (0.2) y(0) = y 0, (0.3) y (0) = z 0 (0.4) Suppose we already know that the functions y 1 (t) and y 2 (t) both solve the homogeneous equation y +by +cy = 0. Suppose we know that y p (t) satisfies the inhomogeneous equation 4

5 (0.2) (but maybe not the initial conditions). What condition on y 1 and y 2 guaratees that the solution of the initial value problem (ODE + initial conditions) must have the form Explain why your condition guarantees (0.5). y(t) = c 1 y 1 (t)+c 2 y 2 (t)+y p (t)? (0.5) 47. Suppose a 2 g mass is attached to a spring, and it is displaced from the rest position by 2cm. The mass is then released. It begins to oscillate, returning to its initial position once every 1/2 second. There is no damping. (i) What is the spring constant k (in units of g/cm)? (ii) Suppose we take the same mass spring system and set it in motion with initial position y(0) = 3 and initial speed y (0) = 1. Find y(t). Start by writing down an initial value problem for y(t). 48. Use variation of parameters to find a particular solution to the equation 49. Suppose y y 2y = 2e t. f(x) = a n (x+3) n n=0 is a given power series. Suppose you know that f(1) converges, but f(10) does not converge. Which of the following must be true? (i) The ROC cannot be less than 3. (ii) The series n=1 na n( 3) n 1 must converge. (iii) The series n=2 n(n 1)a n(14) n 2 does not converge. Justify your answer clearly. 50. Explain why the following series are the same: n= a n (n+1)x n 1 = n= a n+1 (n+2)x n 51. Consider the equation na n x n 1 n n We suppose that a n = 0 for all n 0. 3a n x n = 0 (i) What recurrence relation should a n satisfy in order that this equation hold for all x? Express your answer as a n+1 = (...)a n. (ii) Using this relation, compute a n and evaluate f(x) = n=0 a nx n in terms of a Consider the equation n(n 1)a n x n 2 n We suppose that a n = 0 for all n 0. n 9a n x n = 0 (0.6) 5

6 (i) What recurrence relation should a n satisfy in order that this equation hold for all x? Express your answer as a n+1 = (...)a n +(...)a n 1. (ii) Using this relation, compute a n in terms of a 0 and a 1. (iii) What is the function f(x) = n=0 a nx n (in terms of a 0 and a 1 )? What is the radius of convergence of the series? Hint: from (0.6) you may be able to identify an ODE that f satisfies. 53. Consider the initial value problem y (t)+2y(t) = t, y(0) = 1. (i) Solve the problem using power series y(t) = n=0 a nt n by finding the recurrence relation for a n. (ii) Approximate a solution up to time t = 1 using Euler s method with step size t = 0.01 (using Matlab). Plot the result (on the interval [0, 1]). (iii) Compare the approximate solution from part (ii) with the first four terms in your power series solution. 54. Consider the power series f(x) = ( 1) n7n n 2(x x 0) n n=0 (i) Find the radius of convergence and the interval of convergence. (ii) What is a radius of convergence anyway? Define this concept for a general power series. (iii) What is f (x 0 )? (iv) Find a power series for f (x). What is the radius of convergence? 55. Suppose that y(x) satisfies the initial value problem y (x)+y (x) xy(x) = 0, y(0) = 1, y (0) = 2 (i) Compute the first 5 terms in a power series expansion of the solution around x 0 = 0. (ii) Find a recurrence relation for the coefficients in a power series solution. (iii) What is y (4) (0)? 56. Derive the formula (Lf )(s) = s 2 (Lf)(s) sf(0) f (0). 57. Solve the following initial value problem using the Laplace transform: { y +2y 1, t < 1 +4y = f(t), f(t) = t, t 1 with y(0) = 1, y (0) = Find an initial value problem of the form which has the solution y (t)+py (t)+qy(t) = f(t), t > 0, y(t) = 7e t cos(t)+u 3 (t)e (t 3) sin(t 3) 4 Specifying the initial value problem means finding p,q,f(t),a,b. 6 y(0) = a, y (0) = b t 0 e (t s) sin(t s)ds.

7 59. Solve the following initial value problem using the Laplace transform. When referring to the table of transforms, clearly cite the rules where appropriate (e.g. By rule 2,... ). y +y = sin(t) 2δ(t 4), y(0) = 0, y (0) = Compute the Laplace transform of the function 0, t [0,3] f(x) = 1+t, t [3,4], sin(3(t 4)), t > 4. If using a rule from the table, refer clearly to the rule you are using. 61. Consider the following two statements: (i) If f(x) is a continuous function on [0,1], then the boundary value problem always has at least one solution. y +9y = f(x), x (0,1), y(0) = 0, y(1) = 0 (ii) If f(x) is a continuous function on [0,π], then the boundary value problem always has at least one solution. y +9y = f(x), x (0,π), y(0) = 0, y(π) = 0 One of the statements is true, and one if false. Decide which is true and explain why it is true. For the one that is false, find an example of an f for which there is no solution. 62. Suppose that y 1 satisfies and y 2 satisfies y 1 +p(x)y 1 +q(x)y 1 = f 1, x (a,b), y 1 (a) = 0, y 1(b) = 1 y 2 +p(x)y 2 +q(x)y 2 = f 2, x (a,b), y 2 (a) = 1, y 2(b) = 0. What boundary value problem does the function z(x) = y 1 (x)+y 2 (x) satisfy? 63. What is L 2 ([0,1])? Explain clearly using complete sentences. 64. What does it mean for two functions in L 2 ([0,1]) to be orthogonal? 65. What does it mean for a collection of functions {f n (x)} n=1 L 2 ([0,1])? to be an orthogonal basis for 66. Is it possible to find an infinite set of functions {f n (x)} n=1 in L2 ([0,1]) which are mutually orthogonal but do not form a basis for L 2 ([0,1])? Explain with examples or counterexamples. 67. Your friend claims that for any function f L 2 ([0,1]) there is a sequence of coefficients a 1,a 2,a 3,... such that f(x) = a n sin(nπx). (0.7) n=1 That s rediculous! you say. How about the function: f(x) = 1? Clearly this function is in L 2 [(0,1)]. However, f(0) = 1, while sin(nπ0) = 0 for all n. So, when you add up all the zeros, you have to get zero, not one. Who is right? Explain. It will be useful to reflect on the meaning of equality in (0.7). 7

8 68. If a function f(x) is given as a power series f(x) = n=0 a nx n or as a sine series f(x) = n=1 a nsin(nx), explain why it is useful know only the first few terms in the series. After all, we could never actually evaluate all of the infinitely many coefficients. 69. Suppose you know that a function f(x) is analytic on R. In particular, it is equal to a power series f(x) = a n x n with inifinite radius of convergence. (i) Identify the coefficients a n in terms of f. n=0 (ii) For M > 0 given, explain why f L 2 ([ M,M]). (iii) Are the functions x 2 and x 3 orthogonal in L 2 ([ M,M])? Are the functions x 2 and x 4 orthogonal in L 2 ([ M,M])? Explain. (iv) Identify the coefficients b n in the series representation f(x) = n Z b n e inπx M. in terms of f (not in terms of a n ). Show that b n = b n if f is real-valued. 70. Solve the following eigenvalue problem: y = λy, 2 < x < 5 y (2) = 0, y(5) = 0. (0.8) You may be able to guess the answer, but I want to see that you understand how to derive the answer. In particular, clearly explain the following: why the eigenvalues are real, why there are no other eigenvalues than the ones you find, why the only eigenfunctions are the ones you find (or multiples of them). Be able to answer the same questions for different boundary conditions (dirichlet, neumann, periodic, robin). 71. Fix M > 0. Let f(x) be defined by f(x) = { 0, x [ M,0) 3, x [0,M] Compute the coefficients in the Fourier series expansion of f(x): f(x) = a a n cos n=1 ( nπx ) +b n sin M ( nπx ). M It may be convenient to observe that f(x) 3/2 is an odd function. 8

9 72. Suppose x = (x 1,x 2,...,x d ) R d, then x = d x i e i i=1 and the Euclidean norm of x is Suppose that f(x) L 2 ([0,M]) and Then the L 2 -norm of f is ( d ) 1/2 x = x i 2. (0.9) f(x) = k=1 i=1 ( M f = ( ) kπx c k cos M 0 ) 1/2 f(x) 2 dx Write f in terms of the coefficients c k. The answer will be an infinite-dimensional version of (0.9). 73. Solve the initial value problem 74. Solve the initial value problem u t = 2u xx, x [0,3], (0.10) u(0, x) = cos(3πx) 7 cos(4πx), (0.11) u x (t,0) = 0, u x (t,3) = 0. (0.12) u t = u xx, x [0,1], (0.13) u(0, x) = sin(bπx), (0.14) u(t,0) = 0, u(t,1) = 0, (0.15) where b is a given integer. Notice that the coefficient in front of u xx is negative. So, this is not the heat equation. What happens to u(t,x) as t? What is the role of b? How is this behavior different from the solution of the heat equation u t = u xx with same boundary and initial conditions? 75. Suppose that u satisfies u t = u xx +1, x [0,2], (0.16) u(0,x) = 0, (0.17) u(t,0) = 0, u(t,2) = 0. (0.18) What is the limit ū(x) = lim t u(t,x)? The answer is not ū(x) = 0. Nevertheless, you can figure out the answer without much computation. 9

10 76. Suppose that u satisfies u t = 7u xx, x [0,3], (0.19) u(0,x) = g(x), (0.20) u(t,0) = a, u(t,3) = b. (0.21) where a,b 0. Suppose that l(x) is the unique linear function satisfying l(0) = a, l(3) = b. What initial value problem does the function v(t,x) = u(t,x) l(x) satisfy? 77. Find a simple relation between λ R and b R such that the function u(t,x) = e λt sin(bx) satisfies the heat equation for all x R and t R. 78. Consider the function w(t,x) = 1 e x2 4t 4πt (i) Verify that w(t,x) satisfies the heat equation for all x R, t > 0. (ii) Describe the behavior of this function as t. (iii) Describe the behavior of this function as t For each of the following boundary value problems decide whether there is no solution, infinitely many solutions, or exactly one solution. Justify your answer: (i) (ii) (iii) (iv) y +4y = 0, 0 < x < 2, y(0) = 1,y(2) = 0. y +4y = 0, 0 < x < π, y(0) = 1,y(π) = 0. y 4y = 0, 0 < x < π, y(0) = 1,y(π) = 0. y 4y = 1, 0 < x < π, y(0) = 1,y(π) = 0. Explain, in general, why is it the case that there are either zero, one or infinitely many (rather than exactly 3 solutions, for example)? 80. Make up your own examples (different from the ones above) of boundary value problems where there are zero, one, or infinitely many solutions. 81. Consider the differential operator Ly = y (x) acting on functions of the interval [0,1] which are twice differentiable and satisfy the boundary condition y (0) y(0) = 0 and y (1) = 0. Is the operator self-adjoint? 82. Consider the differential operator Ly = y (x) + xy (x) acting on functions of the interval [0,1] which are twice differentiable and satisfy the boundary condition y(0) = 0 = y(1). Is the operator self-adjoint? 10