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2302f11-mid1-soln - 2(25 Solve the dierential equation dy...

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2. (25) Solve the di ff erential equation dy dx = cos y + 2 xy x sin y - x 2 . Solution. This equation is neither separable nor linear, so we have to hope it is exact. First we express it in the form M dx + N dy = 0: (cos y + 2 xy ) M ( x,y ) dx + ( x 2 - x sin y ) N ( x,y ) dy = 0 . This di ff erential form passes the test for exactness, M/ y = N/ x : M y = - sin y + 2 x, N x = 2 x - sin y, so we know that there should be a function F ( x, y ) such that F/ x = M and F/ y = N , from which it follows that all solutions y ( x ) will satisfy the implicit equation F ( x, y ( x )) = C for all constants C . To find this F we need to undo the partial di ff erentiation: F = M dx = cos y + 2 xy dx = x cos y + x 2 y + g ( y ) , where g is an arbitrary functions of y (but not of x ). To find g , we use the fact that F/ y = N , so F y = - x sin y + x 2 + g ( y ) , and thus g ( y ) = 0. Therefore all solutions y ( x ) satisfy the implicit equation x cos y + x 2 y = C. 2
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3. (25) Make an appropriate substitution in order to find an explicit solution to the di ff er- ential equation dy dx = y x
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