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Campbell Stephanie due 09/12/2010 at 11:55pm EDT. Assignment 1 MATH263, Fall 2010 You may attempt any problem an unlimited number of times. Correct Answers: 1. (1 pt) Match each of the following differential equations with a solution from the list below. y +y = 0 2x2 y + 3xy = y y 12y + 32y = 0 y + 12y + 32y = 0 y = e8x 1 B. y = x C. y = e4x D. y = cos(x) 1. 2. 3. 4. A. Correct Answers: D B C A B E D A 4. (1 pt) Match the following differential equations with their solutions. The symbols A, B, C in the solutions stand for arbitrary constants. You must get all of the answers correct to receive credit. 1. 2. 3. 4. 5. A. B. C. D. E. 2. (1 pt) Match each of the differential equation with its solution. 1. 2. 3. 4. A. B. C. D. y + 13y + 40y = 0 2x2 y + 3xy = y xy y = x2 y +y = 0 y = 3x + x2 y = e8x 1 y = x2 y = sin(x) d2y + 81y = 0 dx2 dy 2xy = dx x2 9y2 dy d2y + 12 + 36y = 0 dx2 dx dy = 18xy dx dy + 18x2 y = 18x2 dx 3 y = Ce6x + 1 3yx2 9y3 = C y = A cos(9x) + B sin(9x) 2 y = Ae9x y = Ae6x + Bxe6x C B E D A Correct Answers: Correct Answers: B C A D 3. (1 pt) Match each differential equation to a function which is a solution. FUNCTIONS A. y = 3x + x2 , B. y = e8x , C. y = sin(x), 1 D. y = x 2 , E. y = 4 exp(6x), DIFFERENTIAL EQUATIONS 1. 2. 3. 4. y + 12y + 32y = 0 y = 6y 2x2 y + 3xy = y xy y = x2 1 5. (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. 5 3 y1 = y1 y2 2 2 3 5 y2 = y1 + y2 2 2 y2 = cos(x) sin(x) A. y1 = sin(x) + cos(x) B. y1 = ex y2 = ex C. y1 = ex y2 = ex 2x D. y1 = 2e y2 = 3e2x 4x E. y1 = e y2 = e4x F. y1 = sin(x) y2 = cos(x) G. y1 = cos(x) y2 = sin(x) -9200 As you can see, nding 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 nd a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions. Correct Answers: CE 9. (1 pt) It is easy to check that for any value of c, the function y = ce2x + ex is solution of equation y + 2y = ex . Find the value of c for which the solution satises the initial condition y(4) = 10. c= Correct Answers: 29754.9817203841 6. (1 pt) It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to nd a function which solves the equation. Two classications are the order of the equation (what is the highest number of derivatives involved) and whether or not the equation is linear . Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear: + sin(t + y) = sin t dt 2 ? 2. y y + t 2 = 0 dy d2y ? 3. t 2 2 + t + 2y = sin t dt dt ? 4. y y + y2 = 0 ? 1. Correct Answers: 2Nonlinear 2Linear 2Linear 2Nonlinear 10. (1 pt) The family of functions y = ce2x + ex is solution of the equation y + 2y = ex . Find the constant c which denes the solution which also satises the initial condition y(5) = 2. c= Correct Answers: -0.0066471471395605 d2y 11. (1 pt) Find the two values of k for which y(x) = ekx is a solution of the differential equation y 12y + 27y = 0. smaller value = larger value = Correct Answers: 3 9 7. (1 pt) It is easy to check that for any value of c, the function c y = x2 + 2 x is solution of equation xy + 2y = 4x2 , (x > 0). Find the value of c for which the solution satises the initial condition y(7) = 1. c= Correct Answers: -2352 12. (1 pt) Find all values of k for which the function y = sin(kt ) satises the differential equation y + 20y = 0. Separate your answers by commas. Correct Answers: 8. (1 pt) The functions y = x2 + are all solutions of equation: xy + 2y = 4x2 , (x > 0). Find the constant c which produces a solution which also satises the initial condition y(10) = 8. c= Correct Answers: 2 c x2 4.47213595499958, -4.47213595499958, 0 13. (1 pt) Find the value of k for which the constant function dx x(t ) = k is a solution of the differential equation 3t 4 3x + dt 4 = 0. Correct Answers: 1.33333333333333 14. (1 pt) Which of the following functions are solutions of the differential equation y 12y + 36y = 0? A. y(x) = 6x B. y(x) = 0 C. y(x) = 6xe6x D. y(x) = x2 e6x E. y(x) = xe6x F. y(x) = e6x G. y(x) = e6x or x. g(y) = E. A curve which at each of its points is perpendicular to the member of the family F that goes through that point is called an orthogonal trajectory to F . Each orthogonal trajectory to F satises the differential equation 1 dy = , dx g(y) where g(y) is the answer to part D. Find a function h(y) such that x = h(y) is the equation of the orthogonal trajectory to F that passes through the point P. h(y) = Correct Answers: 2 * e(6 * (x - 1)) 12 -0.0833333333333333 6*y 1 + (6 /2)*((2 *2 ) - y*y) Correct Answers: BEF 15. (1 pt) Consider the curves in the rst quadrant that have equations y = A exp(6x), where A is a positive constant. Different values of A give different curves. The curves form a family, F . Let P = (1, 2). Let C be the member of the family F that goes through P. A. Let y = f (x) be the equation of C. Find f (x). f (x) = B. Find the slope at P of the tangent to C. slope = C. A curve D is perpendicular to C at P. What is the slope of the tangent to D at the point P? slope = D. Give a formula g(y) for the slope at (x, y) of the member of F that goes through (x, y). The formula should not involve A Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 16. (1 pt) The solution of a certain differential equation is of the form y(t ) = a exp(6t ) + b exp(9t ), where a and b are constants. The solution has initial conditions y(0) = 2 and y (0) = 1. Find the solution by using the initial conditions to get linear equations for a and b. y(t ) = Correct Answers: 5.66666666666667 *(e(6 *t)) + -3.66666666666667 *(e(9 *t) 3 ... View Full Document