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Sum

**Solve the following 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

∴ `((1-y)/y^2)dy = - ((1+x)/x^2)dx`

Integrating on both sides, we get

`int(1/y^2- 1/y) dy = - int (1/x^2+1/x)dx`

∴ `-1/y - log |y| = - (-1/x + 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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