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FE1007 Tutorial 7 March/April 2007 (1) Consider the following first-order ordinary differential equation: 2(2 )xdyyx dxy=+(a) Determine whether the equation is exact. (b) Multiply both sides of the equation by yn, 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) xyen yln sin0xxldxxydyy⎛⎞⎛⎞+++++=⎜⎟⎜⎟⎝⎠⎝⎠(b)()()130ydxxdy−+−=(by two different methods).(c) 2'2xyxyyeyyxe+=−(d) 222211dytydttt=+++Note: 122 222231tan()2()2dxxxcaaxaaxa−⎛⎞=++⎜⎟⎜⎟++⎝⎠∫(3) If the substitution nyvx=transforms the differential equation 23321dyx ydxx y−=into a variable separable equation, determine the value of n. Hence solve the equation.
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