Lesson 2a Separable Equations

Rememberthatseparableequationshavefyythis

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Unformatted text preview: her side of the equation Step 2: Factor out y’. Remember that separable equations have f(y)y’. This means that if it is not multiplication with y’, it is not separable. dy Step 3: Integrate both sides (remember that dx cancels with to make dy). If dx you cannot integrate both sides, you may need to practice your integration techniques . Example 1 Determine which of the following are separable ODEs: y ' xe x x2 f ( y) 1 g ( x) xe x x2 y y' x 1 2 y '( y ' ) 2 x (1 y ' ) 2 x Cannot factor, not separable 1 y' x y' x 1 1 y' x y' x 1 f ( y) 1 f ( y) 1 g ( x) x 1 g ( x) x 1 Example 2 xyy' xy 2 y 2 Find the general solution to: 1y ' ( x 1) y x y' x 1 yxx yy ' xy y x 2 2 y 2 ( x 1) yy ' x yy ' y ( x 1) 2 y xy 2 2 ln | y | x ln | x | c y' 1 1 y x xyy' xy 2 y 2 yy y x x 1 1 dy 1 dx y x e ln| y| e x ln| x| c y' 1 dx 1 dx y x 1 dy 1 dx 1 dx y dx x e xec y ln| x| e Ae x y x Example 3 y ' y 2 1 0 Solve the initial value problem to when y(1)= 1 dy 1dx y2 1 y' y 2 1 y ' 1( y 2 1) ArcTan( y ) x c y' 1( y 2 1) y2 1 y2 1 ArcTan( ) 1 c 0 1 c y' 1 y2 1 y' dx 1dx y2 1 1 dy dx 1dx y 2 1 dx 1 c ArcTan( y ) x 1 Tan( ArcTan( y )) Tan( x 1) y Tan(1 x)...
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