Simon Fraser University MATH 310 Midterm-solutions

MATH 310 Differential Equations Fall 2014 Midterm Exam 31 October 2014 Name: SFU ID: @ Tutorial Section: Student Number: Question 1 2 3 4 5 6 Total Marks Instructions: Show all your work for full credit, and indicate your answers clearly. There are six (6) questions, for a total of 50 points. 1. [8 points] Consider the initial-value problem (1 + x)dy dx = y + (1 + x)2 4 − 3x ; y(1) = −3: Find the longest interval in which the solution to this problem is guaranteed to exist. Then find the solution y(x) of the initial-value problem, and check your solution by substitution into the differential equation. 12. [8 points] Consider the differential equation x dy dx = y + x ey=x: (1) (a) Show that using the substitution v = y=x, equation (1) reduces to the differential equation for v(x) x dv dx = ev: (2) (b) Find the general solution of equation (2), and hence solve the initial-value problem x dy dx = y + x ey=x; y(1) = −2 to find the solution y(x) explicitly. What is the maximal interval of existence of the solution? 23. [10 points] (a) A tank contains 100 litres (L) of water with 6 kg of salt dissolved in it. Pure water is pumped in at 5 L/s, and the mixture is pumped out at the same rate. How long will it take until there is 1 kg of salt in the tank? (b) Consider the differential equation dy dt = (y − 3)2=5 t2 : For which of the following initial conditions can you say for sure (using the Existence-Uniqueness Theorem) that there exists a solution of the corresponding initial-value problem? If a solution is guaranteed to exist, for which initial conditions can you further say for sure that it is the only one? Justify your answers. [Do not attempt to solve the differential equation.] (a) y(0) = 1; (b) y(1) = 2; (c) y(2) = 3: 34. [6 points] Consider the autonomous differential equation (of the form y0 = f(y)) y0 = λ − y2 depending on a parameter λ. For each of the following values of λ, find the critical points (if any) and determine their stability; in addition, for the solution with y(0) = 0, sketch the solution curve and determine the long-time behaviour limt!1 y(t) (without solving the ODE): (a) λ = −1; (b) λ = 1. 5. [6 points] (a) Consider the constant-coefficient differential equation y00 + by0 + cy = 0: (3) Given that y1(t) = e2t and y2(t) = e−t are solutions of equation (3), find the values of the constants b and c. (b) For the values of b and c found in (a), solve the initial-value problem y00 + by0 + cy = 0; y(0) = 6; y0(0) = β: For which value(s) of β does the solution remain bounded as t ! +1? 46. [12 points] (a) Verify that y1(x) = x1=2 is a solution of the linear homogeneous differential equation 4x2y00 + y = 0; x > 0: Use the method of reduction of order or Abel’s formula to find a second independent solution y2(x). (b) Suppose that y1(t) and y2(t) are solutions of the linear homogeneous differential equation y00 + p(t)y0 + q(t)y = 0 on an interval I, where p(t) and q(t) are continuous on I. If y1(t0) = 0 and y2(t0) = 0 for some t0 2 I, prove that y1 and y2 cannot be a fundamental set of solutions on I. (c) Find the general solution of y(4) − 4y000 + 7y00 = 0: 5