# I need help solving this differential equation

Given that,
f(x,y)= x^3 + 3x^2y + 4xy^2 +2y^3

Prove that x df/dx + y df/dy = 3f

I need help at working this out. I have been trying to figure it out all day.

## Answers and Replies

First, are you able to calculate ∂f/∂x, ∂f/∂y and 3f?

Hi,
I am not too sure . I have tried to differentiate it but got mixed up variables.

I differentiated df/dx and got 3x^2 + 6xy +4y^2 and for
df/dy i got 3x^2 + 8xy + 6y^2.
Is this right?

Since f is a function of two variables, x and y, the derivatives of f with respect to x or y are called "partial derivatives", and denoted ∂f/∂x rather than df/dx, for example.

To calculate a partial derivative, then all you have to do is differentiate with respect to the desired variable, viewing all others as constant.

For example: ∂f/∂x = 3x2 + 6xy + 4y2. The "y" is treated as constant in the partial derivative of f with respect to x.

Now, all there is to do is find the partial of f with respect to y and plug them into the equation and see if the equality holds.

I differentiated df/dx and got 3x^2 + 6xy +4y^2 and for
df/dy i got 3x^2 + 8xy + 6y^2.
Is this right?

Looks like we're posting at the same time. Yes, those are what I got as well.

Ok.
I found the partial of f with respect to y. so now i got 2 equations, one with respect to x and one with respect to y. what do you plug into the equations exactly and what is 3f?

Your equation is x (∂f/∂x) + y (∂f/∂y) = 3f. You want to plug your derivatives in for the respective functions in the equation (and distribute the x and y), which will give you the left side of the equation. One the right, plug in the x^3 + 3x^2y + 4xy^2 +2y^3 for f and distribute the 3 across it. Both sides should be equal.

Can you show me what you mean in numbers instead of words because i am having trouble understanding

Sure, you have x ∂f/∂x + y ∂f/∂y = 3f.

Also, ∂f/∂x = 3x2 + 6xy + 4y2 and ∂f/∂y = 3x2 + 8xy + 6y2.

On the left side of the equation:
x(∂f/∂x) + y(∂f/∂y) = x(3x2 + 6xy + 4y2) + y(3x2 + 8xy + 6y2) = 3x3 + 9x2y + 12xy2 + 6y3

On the right side:
3f = 3(x3 + 3x2y + 4xy2 +2y3) = 3x3 + 9x2y + 12xy2 + 6y3.

Both are equal.

Ok.
Thank you for your help!

epenguin
Homework Helper
Gold Member
Isn't this Euler's theorem? (one of them).

The polynomial is homogeneous. That means all the terms are of the form aixiyn-i.

Not difficult to prove that for any homogeneous polynomial of degree n, f = $$\Sigma$$ aixiyn-i

x (∂f/∂x) + y (∂f/∂y) = nf

hey, this is simple eulers theorem application. that is, for a 2 variable fxn,if f(x,y) is a fxn homogeneous of degree n, xdf/dx + ydf/dy = nf(x.y) (note, its all partial derivatives here. so it should be a del operator notation rather than d).

all u need to do is, substitute x=xt and y=yt in the fxn. u will get l.h.s = t^3f(x,y). the degree of t will give the degree of the eqn, if at all homogeneous. and by euler's theorem, the result follows.