Math 24 Final Review Problems August 26, 2015 This is a short list of the types of differential equations we’ve learned to solve throughout this class. This is meant to be an example of the different types of differential equations and techniques you have been shown and not a ‘mock’ exam. The problems are taken from the textbook and some of the solutions can be found in the appropriate solutions pdf on the course website. 2.1.#17 Solve the initial value problem (linear, first order) y 0 − 2y = e2t , y(0) = 2. 2.2.#5 Solve the differential equation (seperable, first order) y 0 = (cos2 x)(cos2 2y). 2.5.#1 Find the equilibrium points, classify each one as stable/unstable, and draw the phase lines for the following equation (autonomous, first order) dy −2(arctan y) = . dt 1 + y2 2.5.#3 Determine whether the following equation is exact. If it is exact, find the solution (3x2 − 2xy + 2) = (6y 2 − x2 + 3)y 0 = 0. 2.8.#4 Using Picard’s method, compute the approximate solutions φn (t) for n = 1, 2, 3, 4 for the following equation y 0 = −y − 1, y(0) = 0. 3.5.#5 Find the general solution to y 00 − 2y 0 − 3y = −3te−t . (second order, linear, nonhomogeneous, repeated roots) note: also know how to do complex roots case and non-repeated real root case. 1 3.8.#6 A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(t/2) N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass. 7.5.#9 Find the general solution to the given system of equations. Draw a direction field and a few of the trajectories. 4 −3 0 x = x. 8 −6 7.7.# 6 Find the fundamental matrix Φ(t) for the given system of equations satisfying Φ(0) = I −1 −4 x0 = x. 1 −1 7.9.#7 Find the general solution to the given system of equations −1 1 1 t x0 = x+ , t > 0. 4 1 2t−1 + 4 5.2.#6 Seek the power series solution to the following differential equation at the point x0 = 0. Find the recurrence relation for the coefficients of the power series. Compute the first four terms of each of the two solutions y1 and y2 . Evaluate the wronskian W (y1 , y2 )(x0 ). If possible, find the general term in each solution (this means solving the recurrence relation). (2 + x2 )y 00 − xy 0 + 4y = 0, 2 x0 = 0.