Question

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

Accepted Answer

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) .

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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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