Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0 - Mathematics and Statistics

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Sum

Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0

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Solution

(x2 – yx2)dy + (y2 + xy2)dx = 0

∴ x2(1 – y) dy + y2(1 + x) dx = 0

∴ x2(1 – y) dy = – y2(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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Chapter 1.8: Differential Equation and Applications - Q.4

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