DIFFERENTIAL EQUATION: INTEGRABALE COMBINATIONS

Show straightforward, complete, and neat solution. Box final answer.

Example of Solution (need to see in the solution) (y dx - x dy) (-x2+ xy - y2) = 0 (y dx - x dy) [xy - (x2 + y2)] = 0 Dividing both sides by [(xy) (x2 + y?)] (y dx - xdy) [xy - (x2 + 2)] = 0 (xy) (x2+ y2) (y dx - x dy) xy - (x2+ yz) ) xy (x2+ y?) = 0 1 (y dx - x dy) = 0 x2 + y2 xy a arctan (#)|- Jain (#) ]= 0 y dx - xdy ydx - x dy = 0 x2+ y2 xy X arctan In d arctan ()- d In () =0 = C - Final answerCommon Integrable Combinations Rule/Derivative Form Rule/Derivative Form Product Rule d(xy) = x dy + ydx Product inside x dy + y dx logarithm d(In xy) = xy d y dx - x dy 41X = y2 d ( In) y dx - x dy = Quotient Rule Quotient inside xy logarithm d x dy - y dx = d ( In !) = xdy - y dx x2 xy Rule/Derivative Form d arctan - y dx - x dy y x2+ y- Quotient inside arctan d arctan - x dy - y dx x2+ yz Chain Rule d(u") = nun-1 du1. ( y dx - x dy ) - 3y dy + y?exdx = 0 2. 2 sin(2xy) (x dy + y dx) = 0 3. (y dx + x dy) (2xyexy) = 0 4. ( x dy - y dx) (x2 + xy + y2) =0 5. y(x dy + y dx) - (xy dx - x dy) = 0