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

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2. (25) Solve the di f 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 Mdx + Ndy =0: (cos y xy ) ± ²³ ´ M ( x,y ) dx +( x 2 - x sin y ) ± ²³ ´ N ( x,y ) dy =0 . This di f erential form passes the test for exactness, M/ y = N/ x : M y = - sin y x, N x =2 x - sin y, so we know that there should be a function F ( x, y )suchthat F/ x = M and 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 ±nd this F we need to undo the partial di f erentiation: F = µ = µ cos y xy dx = x cos y + x 2 y + g ( y ) , where g is an arbitrary functions of y (but not of x ). To ±nd g ,weusethefactthat 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 fnd an explicit solution to the di f er- ential equation dy
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2302f11-mid1-soln - 2. (25) Solve the dierential equation...

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