How to integrate this partial differential equation

In summary, the conversation is about an equation involving partial derivatives and functions of x and y. The question is whether integrating the equation with respect to y will result in a function of x as the constant of integration or just a constant. It is determined that the result will be a function of x, and boundary conditions must be used to find its form.
  • #1
JulieK
50
0
I have the following equation

[itex]\frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0[/itex]

where [itex]y[/itex] is a function of [itex]x[/itex] and [itex]m[/itex] is a function of [itex]y[/itex]. If I integrate this equation first with respect to [itex]y[/itex] should I get a function of [itex]x[/itex] as the constant of integration (say [itex]C\left(x\right)[/itex]) or it is just a constant? If it is a function, how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.
 
Physics news on Phys.org
  • #2
Yes, you should have
[tex]
m(y)\frac{dy}{dx}=C(x)
[/tex]
And therefore you can solve it by
[tex]
m(y)dy=C(x)dx
[/tex]
Which you can integrate.
 
  • #3
JulieK said:
how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.
You'll have to use boundary conditions. There's nothing in the equation that gives a clue about the form of C(x).
 

1. How do I determine the appropriate method for integrating a partial differential equation?

There are several methods for integrating a partial differential equation, including separation of variables, method of characteristics, and finite difference methods. The best method to use depends on the specific equation and its boundary conditions. It is important to carefully examine the equation and its properties before choosing a method.

2. Can I use the same method for all types of partial differential equations?

No, different types of partial differential equations require different methods of integration. For example, linear equations can often be solved using separation of variables, while nonlinear equations may require more complex methods such as finite difference or numerical methods.

3. How can I check if my solution to a partial differential equation is correct?

One way to check the accuracy of your solution is to substitute it back into the original equation and see if it satisfies the equation. Additionally, you can compare your solution to known solutions or use numerical methods to approximate the solution and compare it to your own.

4. Are there any common mistakes to avoid when integrating a partial differential equation?

One common mistake is to overlook boundary conditions or to use incorrect boundary conditions. It is important to carefully consider the boundary conditions and ensure they are included in the solution. Another mistake is to use an incorrect method for the given equation, so it is important to choose the appropriate method based on the equation's properties.

5. Is there a general formula for integrating all types of partial differential equations?

No, there is no one-size-fits-all formula for integrating partial differential equations. Each equation is unique and requires its own method of integration. However, there are general strategies and techniques that can be applied to various types of equations, such as separation of variables and numerical methods.

Similar threads

  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
Replies
5
Views
921
  • Differential Equations
Replies
1
Views
1K
Replies
1
Views
1K
  • Differential Equations
Replies
16
Views
813
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
2
Replies
52
Views
514
  • Differential Equations
Replies
3
Views
1K
Replies
5
Views
1K
  • Differential Equations
Replies
2
Views
915
Back
Top