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Solve the differential equation (x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0
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Solution
(x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0
∴ x^{2}(1 – y) dy + y^{2}(1 + x) dx = 0
∴ x^{2}(1 – y) dy = – y^{2}(1 + x) dx
∴ `((1 - y)/y^2) "d"y = -((1 + x)/x^2) "d"x`
Integrating on both sides, we get
`int ((1 - y)/y^2) "d"y = -int((1 + x)/x^2) "d"x`
∴ `int 1/y^2 "d"y -int 1/y "d"y = -int 1/x^2 "d"x - int 1/x "d"x`
∴ `y^(-1)/(-1) - log|y| = (x^(-1)/(-1)) - log|x| + "c"`
∴ `- 1/y - log|y| = 1/x - log|x| + "c"`
∴ log |x| − log |y| = `1/x + 1/y + "c"`
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