Answer :
We need to show that y = x/(x + c) is a solution of dy/dx = (y - y^2)/x. Then,
dy/dx = ((x + c) * 1 - x * 1)/(x + c)^2
= (x + c - x)/(x + c)^2
= c/(x + c)^2
and
(y - y^2)/x = (x/(x + c) - x^2/(x + c)^2)/x
= (x(x + c) - x^2)/(x(x + c)^2)
= (x^2 + cx - x^2)/(x(x + c)^2)
= cx/(x(x + c)^2)
= c/(x + c)^2
which proves the equality.
dy/dx = ((x + c) * 1 - x * 1)/(x + c)^2
= (x + c - x)/(x + c)^2
= c/(x + c)^2
and
(y - y^2)/x = (x/(x + c) - x^2/(x + c)^2)/x
= (x(x + c) - x^2)/(x(x + c)^2)
= (x^2 + cx - x^2)/(x(x + c)^2)
= cx/(x(x + c)^2)
= c/(x + c)^2
which proves the equality.
We need to show all that solution, so according to the question let us consider the differential equation dy/dx = (y - y^2)/x.
A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable .
How can we prove this equation?
dy/dx = ((x + c) * 1 - x * 1)/(x + c)^2
= (x + c - x)/(x + c)^2
= c/(x + c)^2
So,
(y - y^2)/x = (x/(x + c) - x^2/(x + c)^2)/x
= (x(x + c) - x^2)/(x(x + c)^2)
= (x^2 + cx - x^2)/(x(x + c)^2)
= cx/(x(x + c)^2)
= c/(x + c)^2
Thus, they are equal.
To learn more about differential equations click here:
https://brainly.com/question/25731911