Intro Differential Equations.

Intro Differential Equations. MATH 2066 EL Department of Mathematics and Computer Science LAURENTIAN UNIVERSITY, SUDBURY Deadline: Monday, 19 October 2015 at 9:30 a.m October 4, 2015 EXERCISE 1. [Exact and Non Exact Equations; 24 Points = 6+6+6+6 ] 1. Show that each of the equations in Problems (a) through (b) is exact and solve the given initial value problem. (a) dy dx = - 2xy + y2 + 1 x2 + 2xy ; y(-1) = 2. (b) (yexy cos 2x - 2exy sin 2x + 2x) + (xexy cos 2x - 3)y0 = 0; y(?/4) = 0. 2. Show that the given equation is not exact, find an integrating factor and solve the given equation (x + 2) sin y + (x cos y)y0 = 0, x>0. 3. Show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation. ? sin y y - 2e-x sin x ? + ? cos y + 2e-x cos x y ? y0 = 0, µ(x, y) = yex. EXERCISE 2. [Bernouilli Equations; 24 Points = 7+17] In each of Problem 1 through 2, find the solution of Bernouilli equations 1. y0 = 2 3t ln t y + (ln t)2 1 py , t>0, t 6= 1; y > 0. 2. y0 = t 4(1 - t4) y - 5t 4(1 + t2)2 y-3, y 6= 0; y(0) = 1. 1 EXERCISE 3. [Riccarti Equations; 18 Points= 8+10 ] In each of Problem 1 through 2, solve the Riccarti equations satisfying the initial condition given and where y1 is a particular solution. 1. y0 = (y - t)2 + 1, y1(t) = t; y(0) = 1 2 . 2. y0 = y2 - y x - 1 x2, x>0, y1(x) = 1 x ; y(1) = 2. EXERCISE 4. [Phase line; 16 Points= 8+8 ] Problems 1 through 2 involve equations of the form dy dx = f(y). In each problem sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one asymptotically stable, unstable, or semistable. Draw the phase line in the ty-plane. 1. dy dx = y(y2 - 3y + 2), y0 $ 0. 2. dy dx = y2(1 - y2), -1 < y0 < 1. EXERCISE 5. [18 Points= 4+5+4+5] In each of Problem 1 through 4, find the general solution of the given di?erential equation 1. y00 - y0 - 12y = 0. 2. y000 - y00 - y0 + y = 0. 3. y00 - 2y0 + 5y = 0. 4. y000 + 6y00 + 12y0 + 8y = 0. 2