Sec_2.2 - 2.2 Separable Variables Ex1 Solve dy a 1 e2 x dx...

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2.2 Separable Variables Ex1. Solve a. 2 1 x dy e dx = + Solution : ( 29 ( 29 2 2 2 2 1 1 1 1 . 2 x x x x dy e dy e dx dx dy e dx y x e c = + = + = + = + + b. ( 29 sin dy x dx = Solution : ( 29 ( 29 ( 29 ( 29 sin sin sin cos . dy x dy x dx dx dy x dx y x c = = = = - + Definition 2.1 Separable Variables A differential equation of the form ( 29 ( 29 g x dy dx h y = is said to be separable. Ex2. Solve ( 29 1 0 x dy ydx + - = Solution : ( 29 ( 29 ( 29 ln ln1 ln1 ln1 1 1 1 0 1 1 1 ln ln 1 1 . y x c x x c x dy ydx x dy ydx dy dx dy dx y x y x y x c e e y e e y c e y c x + + + + + - = + = = = + + = + + = = = � � = +
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Ex3. An Initial-Value Problem Solve the initial-value problem ( 29 , 4 3. dy x y dx y = - = Solution : ( 29 2 2 2 2 2 2 1 1 , 4 3 2 2 1 1 25 3 4 2 2 2 25. dy x ydy xdx ydy xdx dx y y x c y c c x y = - = - = - = - + = = - + = + = Ex4. Solve ( 29 sin 0 y x e x dx ydy - - = . Solution : ( 29 ( 29 ( 29 ( 29 ( 29 sin 0 sin sin , ( ) cos sin . y y y y y x e x dx ydy x x dx e ydy x x dx e ydy Integrate by parts x x x ye e c - - = = = - + = - + Ex5. Solve ( 29 4 2 3 2 0 x xy dx y e dy - + + = . Solution : ( 29 ( 29 2 4 2 3 3 4 3 2 4 3 3 2 2 0 1 2 9 6 3 1 . x x x x y xy dx y e dy e xdx dy y e xdx dy y y e x c y y - + + + = = - = - + - = + +
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Ex5. Solve ( 29 2 4, 0 2 dy y y dx = - = - . Solution : ( 29
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