Practice Test 1 on Databases - Spring 2008 | MATH 221, Study notes of Differential Equations

Download Practice Test 1 on Databases - Spring 2008 | MATH 221 and more Study notes Differential Equations in PDF only on Docsity! Math 221 Sec.4 Test I Feb. 8, 2001 Name: Score: Instructions: You must show supporting work to receive full and partial credits. No text book, notes, formula sheets allowed. 1(15 pts) Verify that ln y + xy = 5 defines an implicit solution to the equation dydx = −y2/(1 + xy). 2(18 pts) Find the solution to the initial value problem dydx = x 1+y , y(0) = 1. 3(18 pts) (a) Use the Euler method to approximate the solution to the initial value problem y′ = (x + 1)(y4 + 1), y(0) = 0 at these points x = 0, 0.25, 0.5. (b) Plot your approximating solution in the xy plane. 4(16 pts) Find the general solution to the equation y′ − 1(1+t)y = 1. 5(18 pts) Consider the equation dy dt = f(y) = (1− y)(y − k), where k is a parameter. (a) Sketch a phase line for the equation for some value of k between 0 and 1. Classify each equilibrium point as sink, source, or node. (b) Sketch a few typical solutions qualitatively with y v.s. the time t, for the same value of k you used in (a) above. (c) For what value of k does the equation has only one equilibrium point? Clas- sify that point as sink, source, or node. 6(15 pts) Blood carries a drug into an organ at a rate of 3 cm3/sec and leaves at the same rate. The organ has a liquid volume of 125 cm3 which has no trace of the drug initially. If the concentration of the drug in the blood entering the organ is 0.2 g/cm3, write down an initial value problem for the amount of drug in gram at any time t. (Remark: Don’t think about by-passing differential equations to solve this problem.) END