What is the solution of this differential equation?

In summary, the conversation discusses solving a differential equation with the condition that if a=0, the equation becomes a specific function. The conversation also mentions using the change of variables to simplify the equation.
  • #1
Esmaeil
5
0
how can we solve this differential equation?
(ax+y)∂f/∂x + (ay+x)∂f/∂y =0 with this condition: if a=0 then f(x,y)=x^2 - y^2
 
Physics news on Phys.org
  • #2
Esmaeil said:
how can we solve this differential equation?
(ax+y)∂f/∂x + (ay+x)∂f/∂y =0 with this condition: if a=0 then f(x,y)=x^2 - y^2
[tex]df=\frac{∂f}{∂x}dx+\frac{∂f}{∂y}dy[/tex]

At constant f, [tex]\frac{∂f}{∂x}dx+\frac{∂f}{∂y}dy=0[/tex]

Or equivalently, [tex]\left(\frac{∂y}{∂x}\right)_f=-\frac{(∂f/∂x)}{(∂f/∂y)}[/tex]

Chet
 
  • Like
Likes 1 person
  • #3
The change of variables (in attachement) leads to a very simple PDE :
 

Attachments

  • PDE.JPG
    PDE.JPG
    40.2 KB · Views: 446
  • Like
Likes 1 person

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between an unknown function and its derivatives. It involves the use of derivatives, which represent the rate of change of the function with respect to its independent variables.

2. What does it mean to find the solution of a differential equation?

To find the solution of a differential equation means to determine the specific function that satisfies the equation, given the initial conditions. This function will satisfy the equation for all values of the independent variables.

3. Are there different methods for solving differential equations?

Yes, there are different methods for solving differential equations, such as separation of variables, substitution, and using integrating factors. The method used depends on the type of differential equation and its characteristics.

4. How do initial conditions affect the solution of a differential equation?

Initial conditions are necessary in order to determine the specific solution of a differential equation. They provide the starting values for the independent variables, which are used to find the particular function that satisfies the equation.

5. Can all differential equations be solved analytically?

No, not all differential equations can be solved analytically. Some equations are too complex and have no known analytical solutions. In these cases, numerical methods can be used to approximate the solution.

Similar threads

  • Differential Equations
Replies
2
Views
884
  • Differential Equations
Replies
5
Views
591
  • Differential Equations
2
Replies
52
Views
418
  • Differential Equations
Replies
1
Views
1K
Replies
2
Views
2K
  • Differential Equations
Replies
7
Views
308
  • Differential Equations
Replies
1
Views
609
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
3
Views
1K
Replies
6
Views
2K
Back
Top