Differential Equations, Homogeneous equations

beccajd
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Homework Statement



Use the method for Homogeneous Equations to slove

(xy + y^2) dx - x^2 dy = 0

Homework Equations





The Attempt at a Solution



I attempted to get dx/dy on one side and substitute but could not get farther than this

dx/dy = x^2/(xy + y^2)
 
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beccajd said:

Homework Statement



Use the method for Homogeneous Equations to slove

(xy + y^2) dx - x^2 dy = 0

Homework Equations





The Attempt at a Solution



I attempted to get dx/dy on one side and substitute but could not get farther than this

dx/dy = x^2/(xy + y^2)

Try ##y=ux,\ dy = udx + xdu## to get a separable equation.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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