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Solve `(x^(2) + y^(2))(dy)/(dx) = 2xy` |
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Answer» `(dy)/(dx) = (2xy)/(x^(2) + y^(2))` Set `y = vx`, so that `(dy)/(dx) = v + x(dv)/(dx)` `rArr x(dv)/(dx) = (2v)/(1+v^(2)) - v = (2v - v - v^(3))/(1+ v^(2)) = (v - v^(3))/(1 + v^(2))` `rArr (dx)/(x) = (1 + v^(2))/(v(1-v^(2))) dv` `rArr (dx)/(x) = ((1)/(v) + (1)/(1-v) - (1)/(1+v))dv` (Using partial fractions) Integrating `ln|x| = ln|v| - ln |1-v| - ln|1+v| + ln|k|`, `rArr |x| = |(v)/((1-v)(1+v))||k|, k gt 0` `rArr |x| = |((y)/(x))/(1-(y^(2))/(x^(2)))k|` `rArr |x| = |(xy)/(x^(2)-y^(2))k|` `rArr = ky = +- (x^(2) - y^(2))` where k is the constant of integration. |
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