Math 308 Section 517 Exam 1 – Jean Marie Linhart Spring 2009“An Aggie does not lie, cheat, or steal or tolerate those who do”On my honor as an Aggie, I have neither given nor receivedunauthorized aid on this exam.Printed name:Signature:You may find the following useful:d[arctan(u)]du=11 + u2Recall tan[arctan(u)] = u.• You may use your 4x6 inch notecard for this exam. You must hand it inwith your exam.• You may not use any other notes, a calculator, or your book.• You may not collaborate with your neighbors on this exam.• You must show all appropriate work to receive credit, especially partialcredit.• If you use an integrating factor, identify it!• The instructor will provide additional scratch paper if needed.• Read each question carefully.11) (20 points) Find an implicit solution to the following:dydx= −y3+ 4exy2ex+ 3y222) (15 points) Find an explicit solution to the following:xdydx− y = x3sin(2x) y(π) = 033) (2 points) Identify the dependent and independent variables in:dydθ+yθ= −6θy−2y(1) = 0Independent variable: Dependent variable:(2 points) What kind of a differential equation is this?(15 points) Find an explicit solution to the above initial value problem.44) (10 points) Use separation of variables to find the solution todydx= 15x4(y − 1)2/3y(0) = 1(5 points) Is the solution you found by separation of variables a unique solution?If so, give a theorem stating why. If not, give another solution.55) (5 points) An ice cube (all sides are square) melts with the change in itsvolume proportional to its surface area. Assume it remains cubical as it melts.Initially it is 1 cubic inch in volume, and after 5 minutes it is 1/8 cubic inchin volume. Set up a differential equation representing the change in the lengthof a side of the ice cube as a function of time. Note: a cube has 6 faces that areall perfect squares of the same size.(5 points) Solve the differential equation from above and use the informationgiven to write an explicit equation for the length of a side of the cube as afunction of time.(5 points) Use your solution, above, to calculate when the ice cube completelydisappears.66) (12 points) Find any explicit solution fordydx= 2xy2+ 2x y(√π) = 0This problem is unusual in that there are many different equivalent solutions,i.e. there are many constants that will work. You merely need to find oneconstant that works to make this equation true.77) The following two direction fields are for the differential equationsEquation A:dxdt=t21 + x2Equation B:dxdt=−t21 + x2+ 1(2 points) Identify which direction field goes with which equation.(2 points) Use the correct direction field and solution passing through x(0) = 0to approximate x(2) for equation A.x(2) ≈88) (5 points extra credit) Let y0= F (x, y) be an ordinary differential equationwhere F (x, y) is continuous and differentiable with respect to x and y for all xand y. Let y1(x) and y2(x) be two solutions to this ODE. If y1(0) > y2(0) Is itpossible that y1(1) < y2(1)? Why or why
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