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sf - Math 5525 Final Exam Problems and Solutions Problem...

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Math 5525. May 11, 2010. Final Exam. Problems and Solutions. Problem 1 (10 points) Find the general solution of the problem dy dx = y 2 - x 2 xy + x y . Solution. dy dx = y 2 - x 2 xy + x y = y x , v = y x = v + x dv dx = v, dv dx = 0 , v = C, y = Cx. Problem 2 (15 points) Find the general solution of the equation y x dx + ( y 3 + ln x ) dy = 0 . Solution. This equation has the form Mdx + Ndy = 0 with M y = N x = 1 /x , i.e. it is an exact equation. Therefore, the general solution has the form u ( x, y ) = C , where u x = M, u y = N . u x = M = y x = u = ( x, y ) = y ln x + g ( y ); u y = ln x + g 0 ( y ) = N = y 3 +ln x = g ( y ) = y 4 4 . The general solution is 4 u = 4 y ln x + y 4 = C 1 . Problem 3 (20 points) Find the general solution of the differential equation y 00 + 2 y 0 + y = 1 xe x . Solution. We rewrite the given equation as follows Ly = ( D + 1) 2 y = e - x · 1 x . One can find a particular solution in the form y p = e - x · w ( x ). We have ( D + 1) 2 y p = ( D + 1) 2 ( e - x w ) = e - x · w 00 = e - x · 1 x w 00 = 1 x , w 0 = ln x + c, w = x ln x - x + cx + c 1 . Therefore, the general solution is y = ( c 1 + c 2 x + x ln x ) e - x .
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