How to find critical points for a differential equation?

[CW83]

Homework Statement

f(x,y)= 3x⁵y² - 30x³y² + 60xy² +150

How do I find the critical points where this equation equals zero?

Homework Equations

None?

The Attempt at a Solution

I found the partial derivatives, but I'm stuck now and don't know where to go from here.

dx = 15x⁴y² - 90x²y² + 60y²
dy = 6x⁵ - 60x³y + 10xy + 240

 

[member 428835]

start by setting both equal to 0! then try quadratic factoring for the dx. the dy should be obvious.

 

[member 428835]

also, you ask for critical points and where the equation equals zero...are you sure both occur simultaneously? i haven't checked but don't assume this to be the case

 

[CW83]

Okay so I did that. I got for dx, x = 3 +- sqrt(20)/2
The first part of dy should actually be 6yx^5.

After I have the dx value, I can sub them into the dy equation which gives approximately y = -0.1545 and y = -3.6099. Where do I go from here? There should be 4 separate (x,y) points.

 

[epenguin]

Spell things out a little more carefully. What you quote a solution for I think is really saying

∂f/∂x = 0 at all points along (x/y)² = those numbers, which is actually four lines through the origin I think.

Maybe there is a nice algebraic breakup of your ∂f/∂y but anyway what you are doing should lead you to the four points along the four lines.

Alternative approach is to find (dy/dx)_f and its reciprocal and find the points where both are 0. Well same thing really, -(dy/dx)_f is the ratio of your two derived polynomials.

Also I think the second of yours, what you call "dy" is mistaken and it works out fairly simply.

You might as well have taken the factor 3 out of your starting equation.