Mathematics 20D First Midterm, October 27, 2014 Maximum score on this midterm is 50 points. Write your solutions in the space provided. In case you need more space than provided use the back-page. Important: Show your reasoning! Answers without explanations receive no credit, even if they are correct! Full name: Section: Student ID: Signature: Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Total: 1 2 Problem 1. (10 points: 7+2+1) i) Solve the following dierential equation: cos t , y() = 0. ty + 2y = t ii) Compute limt y(t). iii) What is the maximal time interval on which the solution is dened? 3 Problem 2. (10 points) The size of a colony of bacteria y(t) grows according to the dierential equation: dy = y 2 5y + 6. dt Determine the critical points, the equilibrium solutions and indicate their type, that is stable, unstable or semi-stable. What is the long time behavior of the solution with initial data y(0) = 1? (here discuss whether y(t) is increasing/decreasing and indicate limt y(t); no need to discuss the concavity). 4 Problem 3. i) (10 points= 7+2+1)Solve the following dierential equation: 4y y = 0, y(0) = 2, y (0) = . ii) Find so that the solution approaches zero as t . iii) Find so that the solution approaches zero as t . 5 Problem 4. (8 points: 6+2) Consider the dierential equation: t2 y 3ty + 3y = 0, t > 0. i) Check that y1 (t) = t and y2 (t) = t3 form a fundamental set of solutions (that includes checking that they are solutions!). ii) What is the general solution of the above equation? 6 Problem 5. (12 points=2+10) i) Consider the dierential equation: (3xy + y 2 ) + (x2 + xy)y = 0 Is this equation exact? ii) Solve the equation. Leave the solution in implicit form. HINT: if the answer in part i) is negative, then seek an integrating factor = (x) that makes the equation exact