Portland State University math

MTH 256 W 2024 Exam 1 Instructions: This exam is open note and open book. However, collaboration with other students or consulting resources other than the textbook or class materials is prohibited. To receive full credit, you must show all your work unless otherwise stated. Please provide your answers on the provided template and submit your completed exam as a PDE to gradescope. The exam has 10 problems and will be graded out of 100 points. 1. (10 points total) The direction field for 2𝑥 4𝑦 − 3𝑦 ” is shown below with some curves labeled 𝐴 − 𝐹 are shown below. You do not need to show your work on this problem. 𝑦! = a. (5 points) Which curves are integral curves? b. (5 points) Which curves are solution curves? 2. (10 points) Solve the IVP: 𝑦 ! + 3𝑦 = 𝑒 #$% , 𝑦(0) = 2. 3. (10 points) Find the implicit solutions to the separable ODE 1 + 𝑦” 𝑦! = , 𝑥 Hint: Recall the antiderivative formulas 𝑑𝑡 𝑑𝑡 #& (𝑡) 3 = tan , 3 = ln|𝑡| 1 + 𝑡” 𝑡 4. (10 points) Find implicit solutions to the exact ODE 𝑒 % sin(𝑦) 𝑑𝑥 + (𝑒 % cos(𝑦) + 2𝑦)𝑑𝑦 = 0 of the form 𝐹(𝑥, 𝑦) = 𝑐 for some function 𝐹(𝑥, 𝑦). 5. (10 points) Solve the IVP: 𝑦 ! + 2𝑦 = 𝑦 $ 𝑥 ” , 𝑦(0) = 0. Hint: You may use the following antiderivative 1 3 𝑒 #’% 𝑥 ” 𝑑𝑥 = − 𝑒 #’% (1 + 4𝑥 + 8𝑥 ” ). 32 6. (10 points) Solve the IVP: 𝑦 ! − 𝑦 = −0.1𝑦 ” , 𝑦(0) = 5 7. (10 points) Find implicit solutions to the exact ODE 𝑒 % sin(𝑦) 𝑑𝑥 + (𝑒 % cos(𝑦) + 2𝑦)𝑑𝑦 = 0 of the form 𝐹(𝑥, 𝑦) = 𝑐 for some function 𝐹(𝑥, 𝑦). 8. (10 points) Find an integrating factor 𝜇(𝑥, 𝑦) so that 𝜇(𝑥, 𝑦)(𝑒 % 𝑦)𝑑𝑥 + 𝜇(𝑥, 𝑦)(𝑒 % − 𝑒 % 𝑦 ” )𝑑𝑦 = 0 is an exact equation. You do not need to solve the resulting exact equation. 9. (10 points) Determine all values of (𝑥( , 𝑦( ) such that the initial value problem 1 𝑦! = , 𝑦(𝑥( ) = 𝑦( 2𝑥 − 𝑦 is guaranteed to have at least one solution. Is there a unique solution for the values of (𝑥( , 𝑦( ) for which the existence of a solution is guaranteed? You must compute all quantities necessary to apply theorem 2.3.1 to receive full credit but you do not need to specify the open rectangle? 10. (10 points total) Determine if the following ODEs are exact or not. You must justify your answer to obtain full credit. Do not solve the ODEs. a) 6𝑥 ” 𝑦𝑑𝑥 + 2𝑥 $ 𝑑𝑦 = 0 b) (3𝑥 ‘ 𝑦 $ + 𝑥 ” )𝑑𝑥 + (3𝑥 ‘ 𝑦 $ + 𝑦 $ )𝑑𝑦 = 0 c) (cos(𝑥) + 4𝑥 $ 𝑦 ” )𝑑𝑥 + (sin(𝑦) + 2𝑥 ‘ 𝑦)𝑑𝑦 = 0 d) 𝑦𝑒 %) 𝑑𝑥 + 𝑥𝑒 %) 𝑑𝑦 = 0