# 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?

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

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

Gold Member
also, you ask for critical points and where the equation equals zero...are you sure both occur simultaneously? i havent checked but dont 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.

Homework Helper
Gold Member
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.

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