classify the equation: (x)dy/dx + xy = 1-y linear, nonlinear, separable,exact, homogeneous, or one that requires an integration factor?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the exact differential equation, hence you should  divide by x both sides to preserve the equation such that:

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

You need to keep `dy/dx`  to the left side, hence you need to subtract y both sides:

`(dy)/dx = (1-y)/x - y =gt dy/dx = (1-y - xy)/x`

You need to multiply by dx both sides such that:

`dy = ((1-y - xy)dx)/x`

You need to divide by 1-y-xy both sides such that:

`(dy)/(1-y - xy) = (dx)/x =gt (dy)/(1-y - xy)- (dx)/x = 0`

You need to come up with the notation `P(x,y) = 1/(1-y - xy).` Integrating P(x,y) with respect to y yields:

`int (-dy)/(y(x+1)-1) = (-1)/(x+1)*int (dy)/(y - 1/(x+1)) = (-ln|y-1/(x+1)|)/(x+1) + c`

You need to come up with the notation `Q(x,y) = 1/x.`

Integrating Q(x,y) with respect to x yields:

`int (dx)/x = ln|x| + c`

Considering the terms from both integrals yields:

`f(x,y) = 1/x -(ln|y-1/(x+1)|)/(x+1)`

The solution to the exact differential equation is  `(x+1)/x + c= ln|y - 1/(x+1)| =gt y = e^[(x+1)/x + c] + 1/(x+1)` .