Net ID: Name:Midterm #1MATH 286 — Differential Equatio n s PlusThursday, February 13• No notes, personal aids o r calculators are permitted.• Answer all questions in the space provided. If you require more space to write your answer, you may continueon the back of the page. There is a blank page at the end of the exam for r ough work.• Explain your work! Little or no points will be given for a correct answer with no explanation of how you got it.Good luck!Problem 1. (5 points) Circle the slope field below which belongs to the differential equation exy′= x − y.-3-2-10123-3-2-10123-3-2-10123-3-2-10123-3-2-10123-3-2-10123Problem 2. (20 points) Find the general s olution to the differential equation y(5)−4y(4)+ 4y(3)= 0.y(x) =Michael Brannan Armin [email protected] [email protected]/4Problem 3. (20 points) Solve the initial value problem(x2+ 1)dydx+ xy =1x2+ 1√, y(0) = 1.y(x) =Problem 4 . (20 points) The time rate of change of a rabbit population P is proportional to the square root of P .At time t = 0, the population numbe rs 100 rabbits and is increasing at the rate of 20 rab bits per month. How manyrabbits will there be after two months?P (2 months) =Michael Brannan Armin [email protected] [email protected]/4Problem 5. (20 points) For each c > 0, let yc(x) =x3, if x < 0,0, if 0 6 x 6 c,(x −c)3, if x > c.(a) Sketch the graph of yc(x) for c = 3/2.(b) Show that, for all c > 0, ycis a solution to the initial value problemdydx= 3y2/3, y(0) = 0.(c) Explain why (b) does not contradict the theo rem on existence and uniqueness for solutions to initial valueproblems.Michael Brannan Armin [email protected] [email protected]/4Problem 6. (20 points) Find a gene ral solution to the differential equationx2dydx−x2− y2−3xy = 0.y(x) =Problem 7. (5 points) Consider the differential equation Hint: Do not attempt to solve the DE.y′= y4+ x4+ 1.Is it possible that there exists a soluti on with the property that limx→ ∞y(x) = −∞? Why, or why not?Michael Brannan Armin [email protected]
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