I solving this differential equation

In summary, euler's theorem states that for a 2 variable fxn homogeneous of degree n, xdf/dx + ydf/dy = nf(x.y).
  • #1
nikk834
27
0
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.
 
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  • #2
First, are you able to calculate ∂f/∂x, ∂f/∂y and 3f?
 
  • #3
Hi,
I am not too sure . I have tried to differentiate it but got mixed up variables.
 
  • #4
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?
 
  • #5
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. :smile:
 
  • #6
nikk834 said:
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.
 
  • #7
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?
 
  • #8
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.
 
  • #9
Can you show me what you mean in numbers instead of words because i am having trouble understanding
 
  • #10
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.
 
  • #11
Ok.
Thank you for your help!
 
  • #12
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 = [tex]\Sigma[/tex] aixiyn-i

x (∂f/∂x) + y (∂f/∂y) = nf
 
  • #13
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.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is used to model real-world phenomena in fields such as physics, chemistry, and engineering.

2. Why is solving differential equations important?

Solving differential equations allows us to predict and understand the behavior of systems and processes in various fields. It is also crucial in developing mathematical models and making accurate predictions in scientific research.

3. What are the methods for solving differential equations?

There are various methods for solving differential equations, including separation of variables, substitution, variation of parameters, and numerical methods such as Euler's method and Runge-Kutta method.

4. How do you know which method to use for solving a specific differential equation?

The method used to solve a differential equation depends on its type and characteristics. For example, a separable differential equation can be solved using separation of variables, while a non-linear differential equation may require numerical methods.

5. Can differential equations be solved using software?

Yes, there are many software programs, such as MATLAB and Wolfram Mathematica, that can solve differential equations numerically. These programs use algorithms and computational methods to find approximate solutions to differential equations.

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