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Chapter Notes (2)

18 2 differential equations example 24 determine

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Unformatted text preview: ifferential Equations Example 2.4: Determine which of the following differential equations is separable. a) dy = xy + x dx b) dy y + 1 = dx x + 1 c) dy = y ( y + 1) dx d) dy = x+ y dx Solution: a) The right side may be factored as x( y + 1) , which meets the condition for separability. b) The right side is the quotient of a function of y divided by a function of x . Therefore, this equation is separable. c) The right side can be thought of as a product y ( y + 1) × 1 , where the constant factor 1 can be viewed as a function of x . Therefore, the equation is separable. d) The right side cannot be factored as a product of a function of x and a function of y . Therefore, this equation is not separable.! The idea behind the method of separation of variables is to multiply or divide both sides of the separable differential equation by suitable factors so that all the y terms wind up on the left side, together with the unknown derivative, and all the x terms wind up on the right side. Following that step, we then have two big hurdles to overcome. First, we must be able to integrate both sides of the transformed equation. Second, even after integrating, we will not usually have an explicit formula for the solution but rather an implicit one. Some tricky algebra may be needed to obtain an explicit solution. Often, an explicit solution cannot be found. We illustrate with some examples, beginning with a reprise of Example 2.2. Example 2.5: Use the method of separation of variables to find the general solution of dy = y. dx Solution: ! Although we will eventually work with the differential notation, to understand the method it is better to write the equation as y′ = y . Following the instructions in the previous paragraph, we divide both side of the equation by y . (It is worth emphasizing that division and/or multiplication must be used to move factors from one side to the other, never addition or subtraction.) We then obtain y′ = 1. y 19 2 Differential Equations Now we integrate both sides of the latter equation with respect to x , the independent variable. This gives y...
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