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Unformatted text preview: McGill University Math 325B: Differential Equations Notes for Lecture 3 Text: Section 2.6 In this lecture we will treat Bernoulli and homogeneous ODEs by means of a change of dependent variable to reduce the Bernoulli ODE to a linear one and the Homogeneous ODE to a separable one. Bernoulli Equations. A Bernoulli ODE is a differential equation of the form y = p ( x ) y + q ( x ) y n , where n 6 = 0 , 1. Notice that if n > 0, the zero function y = 0 is a solution. So we assume that y is not the zero function. In order to transform this DE to a linear one we first divide both sides of the equation by y to get y y n = p ( x ) y n- 1 + q ( x ) . Setting u = 1 /y n- 1 and using the fact that du dx = (1- n ) y /y n , the given ODE can be written du dx = (1- n ) p ( x ) u + (1- n ) q ( x ) , which is a linear equation. Example. The ODE y = y- y 2 is a Bernoulli equation. It is also a separable equation. We will solve it first as a Bernoulli equation and then as a separable equation and compare the two methods....
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