Math 285 Final Exam Practice: ODEs, BVPs, Fourier Series, Heat, Population Dynamics, Exams of Differential Equations

Download Math 285 Final Exam Practice: ODEs, BVPs, Fourier Series, Heat, Population Dynamics and more Exams Differential Equations in PDF only on Docsity! NAME: Math 285 — Final exam practice Total points: 100. Please show the work you did to get the answers. Calculators, computers, books and notes are not allowed. Suggestion: even if you cannot complete a problem, write out the part of the solution you know. You can get partial credit for it. 1. [10 points] Find the general solution of the following ODE for y(x): y(5) − 4y(4) + 4y(3) = 0 (where y(4) means the fourth derivative of y) 1 NAME: 2. [10 points] Solve the following initial value problem for y(x): y′′ − 1 x2 = 0; y(1) = y′(1) = 0 2 NAME: 5. [10 points] Calculate the Fourier series expansion for the following func- tion of period 2: f(t) = 2 + 2 t2 for − 1 < t < 1 5 NAME: 6. [15 points] Find the solution y(x, t) in 0 < x < 3 and t ≥ 0 for the following heat conduction problem: 2yt = yxx; yx(0, t) = yx(3, t) = 0; y(x, 0) = g(x) where g(x) is a generic function. Derive your solution using separation of variables. Don’t rely on a formula. What happens as t → +∞? Briefly discuss the physical meaning of this result. 6 NAME: 7. [10 points] Consider the following population equation for P (t) (assume it makes sense to also consider negative values of P here). Sketch a slope field and indicate on it the equilibrium solutions and their stability. Then sketch the behaviour of the three solutions corresponding to the given initial conditions (which are not given at t = 0!). dP dt = 16P − 8P 2 + P 3 a) P (1) = −1 b) P (1) = 1 c) P (1) = 5 7