Math 251October 12, 2005 First ExamNAME: Section #:There are 9 questions on this exam. Many of them have multiple parts. The p oint value of each questionis indicated either at the beginning of each question or at the beginning of each part where there aremultiple partShow all your work. Partial credit may be given.The use of calculators, books, or notes is not permitted on this exam.Please turn off your cell phone before starting this exam.Time limit 1 hour and 15 minutes.Question Score1 22pt2 10pt3 10pt4 8pt5 12pt6 10pt7 10pt8 8pt9 10ptTotal 100pt1. a. 2pt Consider the following differential equation y0= y + 2t. Without solving it, determine the slope ofthe tangent line to the solution at the point (1, 2).b. 2pt Find the Wronskian W (y1, y2) of the functions y1= sin t and y2= cos tc. 2pt Supp ose y1and y2are two solutions of the ODE y00+ (sin t)y0+ y = 0. and suppose thattheir Wronskian by W (y1, y2)(t) is 2 at t = 0. Find W (y1, y2)(t) for any t.For the initial value problems in parts d. through g. state whether or not one of our two existence and uniquenesstheorems for first order ODE’s guarantees a unique solution. If the answer is yes and the theorem provides an intervalof existence, then state what the interval is without actually solving the equation.d. 2pt (t + 2)y0+ (y − 1)2/3= 0, y (3) = 0e. 2pt (t + 2)y0+ (y − 1)2/3= 0, y (0) = 1f. 2pt (t2+ 2t)y0+ y = 0, y(−3) = 0g. 2pt (t2+ 2t)y0+ y = 0, y(3) = 0In parts h. 2pt through j. assume that L[y] = y00+ p(t)y0+ q(t)y and that p(t) and q(t) are continuous functions onthe entire real axis (−∞, ∞).h. 2pt Circle a pair of functions among the following functions which could be a fundamental setof solutions for the differential equation L[y] = 0. (There is more than one correct answer.)y1= e3+t/2, y2= e3+2t, y3= e2t−2, y4= 0j. 2pt Suppose that y1and y2have the following properties: L[y1] = t, and L[y2] = 0. Then oneof the following solves the differential equation L[y] = 2t. Circle it.y1/2 + y22y1+ 2y22y2y1+ y2In parts k. through m. match the ODE’s on the left with a description on the right.k. 2pt y0= ey +tl. 2pt y0= 2 − ym. 2pt y0= t2− 3yi. linear and separableii. linear but not separableiii. separable but not lineariv. not separable and not linear2. Consider the autonomous differential equationy0= (y − 6)(2 − y) = −y2+ 8y − 12a. 2pts Sketch a direction field for this equation. Indicate the equilibrium solutions in your sketch.b. 2pts Which equilibrium solution is (are) asymptotically stable and which is (are) unstable.c. 3pts Find a formula for y00in terms of y.e. 3pts Sketch the graph of the solution with t ≥ 0 with the initial value y(0) = 3 indicating itsconcavity as accurately as possible.3. 10 pt Find the general solution to the differential equationty0= t6+ 5y t > 04 a. 6pts Find the general solution to the following IVP3y0= 4y4t3b. 2pts Find the solution to the above differential equation that satisfies the initial conditiony(1) = 25. a. 6pt Find the general solution the following differential equation with initial conditions:y00+ 6y0+ 13y = 0b. 4pt Solve the following IVP for the ab ove ODE:y(0) = 1, y0(0) = 3c. 2pt Solve the following IVP for the above ODE:y(999) = 1, y0(999) = 36. a. 2pt One of the following differential equations is exact. Circle it:2t + 3y + (2y + 3t)y0= 0 2t + 3y + (2t + 3y)y0= 0b. 8pt Find the solution to the following differential equation which satisfies y(1) = 0:y2+ t2+ (e2y+ 2ty)y0= 07. A tank with a capacity of 200 liters initially contains a mixture of 25 grams of salt disolved in 50liters of water. A salt water mixture with a concentration of 2 grams/liter enters the tank at therate of 6 liters/min. Well stirred mixture leaves the tank at 6 liters/min.a. 4pt Let Q(t) be the quantity of salt in the tank at time t ≥ 0. Write down a differentialequation and an initial condition for the quantity Q(t) of salt in grams in the tank at any timet ≥ 0. (Do not solve it.)b. 2pt Draw a direction field for the ODE.c. 2pt Without solving the ODE, determine approximately the quantity of salt in the tank aftera long time.d. 2pt Without solving the ODE, determine Q0(0).8. Match the differential equations listed below with the descriptions of long time behavior listed below.(Each description matches only one equation. Please place your answer in the space provided.)a. 2pt y00− y0− 2y = 0b. 2pt y00+ 4y0+ 4y = 0c. 2pt y00− 4y0+ 29y = 0d. 2pt y00+ y = 0I. Every solution approaches 0 as t → ∞II. Has a nonzero solution that approaches 0as t → ∞ and has a nonzero solution that ap-proaches ∞ as t → ∞III. Every nonzero solution approaches either ∞or −∞ as t → ∞IV. Every nonzero solution has oscillationswhich become progressively larger as t → ∞V. Every nonzero solution has oscillations whichbecome progressively smaller as t → ∞VI. Every nonzero solution oscillates with con-stant amplitude t → ∞.9. 10pt The ODEt2y00− ty0+ y = 0, t > 0obviously has a solution y1= t. Use the method of reduction of order to find another solution ofthis linear homogeneous ODE that is not a constant multiple of
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