Referencing Styles : Not Selected MXB106 Take Home Assignment 3 Question 1 Verify that the piecewise-defined function y(x) = ( ?x 2 x < 0 x 2 x ? 0 is a solution of the differential equation xy0 ? 2y = 0 where x ? (??, ?). Question 2 Consider the first order ODE: dy dx = (y ? 1) 2 + ?1 where y(0) = 1 + ?2 for constants ?1 and ?2. Find the general solution to this problem, then plot the solution over the domain x ? [0, 1] for &ep ... View More MXB106 Take Home Assignment 3 Question 1 Verify that the piecewise-defined function y(x) = ( ?x 2 x < 0 x 2 x ? 0 is a solution of the differential equation xy0 ? 2y = 0 where x ? (??, ?). Question 2 Consider the first order ODE: dy dx = (y ? 1) 2 + ?1 where y(0) = 1 + ?2 for constants ?1 and ?2. Find the general solution to this problem, then plot the solution over the domain x ? [0, 1] for ?1 = ?2 = 0, ?1 = 0 and ?2 = 0.01, and ?1 = 0.01 and ?2 = 0. Comment on how small changes in ?1 and ?2 impact the solution. Question 3 Solve ( ? x + x) dy dx = ? y + y Question 4 The solution of the differential equation 2xy (x 2 + y 2) 2 dx + 1 + y 2 ? x 2 (x 2 + y 2) 2 dy = 0 is a family of curves that can be interpreted as streamlines of fluid flow around a circular object whose boundary is described by the equation x 2 + y 2 = 1. Find the implicit solution of the ODE. 1 Question 5 In the study of population dynamics, one of the most famous models for a growing but bounded population is the logistic equation dp dt = p(a ? bp) where a and b are positive constants. Solve this ODE using the fact that is a Bernoulli equation. Question 6 Suppose an RC series circuit has a variable resistor. If the resistance at time t is given by R(t) = k1 + k2t where k1, k2, and C are positive constants then R dq dt + 1 C q = E(t) If E(t) = E0 and q(0) = q0, where E0 and q0 are constants, show that q(t) = E0C + (q0 ? E0C) k1 k1 + k2t 1 Ck2 Question 7 Given that y1(x) = 1 is a solution to the ODE (1 ? x 2 )y 00 + 2xy0 = 0 use reduction of order to find a second solution y2(x). Question 8 (a) Solve y 00 ? 2y 0 + 2y = 0 where y(0) = 0 and y 0 (?) = 1. (b) Solve y 00 + 4y 0 + 5y = 35e ?4x where y(0) = 1 and y 0 (0) = 0. 2 Question 9 Solve the in-homogeneous ODE y 00 ? 4y 0 + 8y = (2x 2 ? 3x)e 2x cos(2x) + (10x 2 ? x ? 1)e 2x sin(2x) Question 10 (a) Construct a linear, first order ODE of the form xy0 + a0(x)y = g(x) for which yc = c x 3 , and yp = x 3 such that the general solution is y = x 3 + c x 3 (b) Find a condition of the form y(x0) = y0 for the ODE in part (a) so that the solution is y = x 3 ? 1 x 3 3 Read Less