FE1007_Tutorial_7.pdf - FE1007 Tutorial 7 March/April...

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FE1007 Tutorial 7 March/April 2007 (1) Consider the following first-order ordinary differential equation: 2 ( 2 ) x dy y x dx y = + (a) Determine whether the equation is exact. (b) Multiply both sides of the equation by y n , where n is an integer. Determine the value of n if the resulting differential equation is exact. (c) Solve the exact differential equation obtained in part (b). (2) Solve the following ODEs: (a) x y e n y ln sin 0 x x l dx x y dy y + + + + + = (b) ( ) ( ) 1 3 0 y dx x dy + = (by two different methods). (c) 2 ' 2 xy xy ye y y xe + = (d) 2 2 2 2 1 1 dy t y dt t t = + + + Note: 1 2 2 2 2 2 2 3 1 tan ( ) 2 ( ) 2 dx x x c a a x a a x a = + + + + (3) If the substitution n y vx = transforms the differential equation 2 3 3 2 1 dy x y dx x y = into a variable separable equation, determine the value of n. Hence solve the equation.
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