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Consider the differential equation dy/dx = (y - y^2)/x.


Verify that y = x/(x + C) is a general solution for the given differential equation and show that all solutions contain (0,0). ...?

Verified 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. 

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

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