# Thread: A tricky ODE and finding extrema questions!

1. ## A tricky ODE and finding extrema questions!

Hi,

my name is Dimos and I am coming from Greece.

Right to the point.

1) Can somebody help me with the:

$(2x*cos(y)+y*cos(x))+(-x^2*sin(y)+x)*dy/dx=0$

ODE?

I ve tried to find a integration factor and change variable (e.g. v=2x*cos(y) ). I have no clue! Is this ODE a suitable example for polytechnics student?

2) Is there any higher order derivative test to find extrema of multi variable functions? Where can i find more details?

3) Can the use of Lagrange multipliers reveal the nature of extrema (i.e. is it a local minimum, local maximum or saddle point)? How?

Thank you in advance for any help!

Dimos
Athens.

2. Just a reality check. It is SO CLOSE to "exact" that I am first inclined to make sure there is not just a typo.

3. Originally Posted by TKHunny
Just a reality check. It is SO CLOSE to "exact" that I am first inclined to make sure there is not just a typo.
I thought it was a typo. If x in the dy/dx term was sin(x), the ODE would be exact and having the solution

$x^2*cos(y)+y*sin(x)=c$

the person gave it to me for solving is not available at the moment. So I assumed it s correct.

Anyway it may be an interesting (and hard to solve) ODE.

Thanks for your help TKHunny!

Dimos

4. Originally Posted by ouranosky
2) Is there any higher order derivative test to find extrema of multi variable functions? Where can i find more details?
Do you know about the second partials test?

THEOREM: Second Partials Test.

Let $f$ be a function of two variables with continuous second order partial derivatives in some disk centered at a critical point $(x_0,y_0)$, and let

$D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f_{xy}^2$

(a) If $D>0$ and $f_{xx}(x_0,y_0)>0$, then $f$ has a relative minimum at $(x_0,y_0)$

(b) If $D>0$ and $f_{xx}(x_0,y_0)<0$, then $f$ has a relative maximum at $(x_0,y_0)$

(c) If $D<0$ , then $f$ has a saddle point at $(x_0,y_0)$

(d) If $D=0$, then no conclusion can be drawn
Source: Calculus : Multivariable by Anton, Bivens, Davis. 8th Ed.

3) Can the use of Lagrange multipliers reveal the nature of extrema (i.e. is it a local minimum, local maximum or saddle point)? How?
Yes you can show these things. I don't have the time to go through the process, but you need to use this equation(s):

$\nabla f(x_0,y_0)=\lambda\nabla g(x_0,y_0)$ <---Use when you have two variables and one constraint.

$\nabla f(x_0,y_0,z_0)=\lambda\nabla g(x_0,y_0,z_0)$ <---Use when you have three variables and one constraint.

where $\lambda$ is the Lagrange Multiplier.

I hope this helps!

--Chris