Differential Equation - Elementary Differential Equations - Exam, Exams of Differential Equations

Download Differential Equation - Elementary Differential Equations - Exam and more Exams Differential Equations in PDF only on Docsity! Math B21 Final December 11, 1996 C. Robinson 1. (20 Points) Find the general solution (implicit if necessary, explicit if convenient) of dy dx = 3 √ xy. 2. (25 Points) Consider the differential equation dy dt = 3y − y2. (a) Determine all the equilibria (critical points) and their stability or instability. (b) Determine where 3y − y2 is positive and where it is negative. (c) Use the above information to construct a sketch of typical solutions. Plot your solutions in (t, y) space. (You do not need to include the slope field or find the explicit solutions.) 3. (25 Points) Find the general solution of ÿ + 2ẏ + 5y = sin(2t) 4. (25 Points) Apply the improved Euler method with step size h = 0.1 and initial conditions y(0) = 2 to the system of equations dy dt = ty to calculate an approximation for y(0.2) 5. (21 Points) Consider the three equations (i) xy′′ + (x+ 3)y′ + (1− x2)y = 0 (ii) x2y′′ + (x+ 3)y′ + (1− x2)y = 0 (iii) xy′′ + x2y′ + x4y = 0 (a) For which of the three equations above is x = 0 an ordinary point, a regular singular point, or an irregular singular point? (b) Also indicate the form of the solution that is appropriate in each case. 6. (9 Points) The indicial equation for the equation 2x2y′′ + 5xy′ + (x− 2)y = 0 is 2r2 + 3r− 2 = 0. Is either solution defined at x = 0? For which r? 7. (25 Points) Consider the solution of the equation (1− x2)y′′ − 2xy′ + 6y = 0 near x = 0. (a) Find the recurrence relation for the coefficients. (b) Find the first three terms of the solution with y(0) = 0 and y′(0) = 1. 1