Verify that (x^2)y+y^2 =c is a solution of 2xydx+(x^2+y)dy=0

Question

Differential equations question

Answer 1

Hey Pheonix.

 

If

 

(2xy)dx + (x²+y)dy = 0

 

that implies that:

 

dy/dx = (-2xy)/(x²+y)

 

If we simply wish to verify that x²y + y² = c is a solution, then we just need to differentiate that equation and see if it matches our conclusion above. We'll have to differentiate implicitly.

 

d/dx[x²y + y² = c]

 

= 2xy + x²*dy/dx + 2y*dy/dx = 0

 

Subtract 2xy from each side.

 

x²*dy/dx + 2y*dy/dx = -2xy

 

Factor out dy/dx

 

dy/dx(x² + 2y) = -2xy

 

dy/dx = (-2xy)/(x² + 2y)

 

Either I screwed up somewhere, or it turns out it isn't a solution. Are you sure you copied it down correctly?