Page 1 of 7Student Name:Student Net ID:MATH 286 SECTION X1 – Introduction to Differential Equations PlusMIDTERM EXAMINATION 1September 25, 2013INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam 60 minutes long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Student Net ID: MATH 286 X1 Page 2 of 7Question: 1 2 3 4 5 TotalPoints: 6 9 10 8 7 40Score:1. (6 points) Solve the initial value problemdydx= y−1eycos x, y(0) = 1.An implicit expression for the solution is fine.Student Net ID: MATH 286 X1 Page 3 of 72. Consider the ordinary differential equationdydx=yx+ x ln x (x > 0).(a) (3 points) Without explicitly solving this ODE, determine whether the correspond-ing initial value problem with initial condition y(1) = 0 has a unique solution, nosolution, or more than one solution. Explain your answer!(b) (6 points) Find the solution to the IVPdydx=yx+ x ln x, y(1) = 1.Student Net ID: MATH 286 X1 Page 4 of 73. Consider the ODExx2+ y2+ 1− sin x +yx2+ y2+ 1dydx= 0 ((x, y) 6= (0, 0)).(a) (4 points) Show that this ODE is exact.(b) (6 points) Find an implicit expression for the general solution to this ODE.Student Net ID: MATH 286 X1 Page 5 of 74. (8 points) Find the general solution to the second order ODExy00= x expy0x+ xy0(x > 0),where exp a = ea. (Note: It is OK if your final answer involves an indefinite integralthat cannot be evaluated).Student Net ID: MATH 286 X1 Page 6 of 75. (7 points) The functionsy1(x) = ex, y2(x) = e−x, y3(x) = e2x(x ∈ R),are solutions to the 3rd order linear ODEy000+ Ay00+ By0+ Cy = 0.What are A, B, C?Student Net ID: MATH 286 X1 Page 7 of 7(Extra work
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