Ordinary Differential Equations — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1. (a) Determine the constant A such that the differential equation is exact, and hence solve the resulting equation (b) Solve the initial-value problem (c) A cake is removed from an oven at 210°F and left to cool at room temperature, which is 70°F. After 30 minutes, the temperature of the cake is 140°F. In how much time will the cake's temperature reach 100°F? 2. (a) In a certain culture of bacteria, the number of bacteria increased sixfold in 10 h. Assuming the bacteria continue to grow exponentially at a constant rate, how long did it take for the population to double? (b) Solve the differential equation by reducing the order xy" + y' = 4x. (c) The world's total population reached 6 billion persons in mid-1999, and was then increasing at the rate of about 212 thousand persons each day. Assuming that natural population growth at this rate continues, then (i) What is the annual growth rate k? (il) What will be the world population in mid-2050? 3. (a) Show that e* and eare linearly independent solutions of the differential equation y" - 3y' + 2y = 0. What is the general solution? Find the particular solution satisfying the initial conditions y(0) = 1, y'(0) = 1. (b) Solve the differential equation using the method of variation of parameters. (c) Find the general solution of the differential equation using the method of undetermined coefficients. 4. (a) Use the operator method to find the general solution of the following linear system : (b) Find the general solution of the differential equation, assuming that x > 0 (c) A body with mass 2 kg is attached to the end of a spring that is stretched 2 m by a force of 100 N. It is set in motion with an initial position X, = Im and initial velocity Vo = -5 m/s. Find the amplitude, period, and frequency of the resulting motion. (a) Derive the epidemic model of influenza dR = yI where B and y are positive constants, S(t), I(t), R(t) denote the number of susceptibles, the number of infectives and the number of recovered at time t respectively. (b) Define equilibrium points and find the equilibrium solutions of the Predator- Prey model : ax = BIX -qXY, dy dt at =CZXY-azY, where X and Y denote prey and predator density, respectively. Positive constants c,, C, are interaction parameters, B, is the per capita birth rate of prey and a, is the per capita death rate of the predator. Also, draw a sketch of the trajectory direction in the phase plane. (c) The fish population P(t) in a fish farm, follows a density-dependent growth model given by the logistic equation di =0,5 (1 - 2000) where t is measured in months, and initially at t = 0 the pond is stocked with (i) Identify the intrinsic growth rate (r) and the carrying capacity (K) of the pond from the equation. (i) Solve the logistic equation to find the expression for P(1). (iii) Calculate the fish population after 10 months. 6. (a) Consider the following differential equation of a lake pollution model.