Seperation of variables - first order PDE

In summary, separation of variables is a technique used to solve first order partial differential equations (PDEs) by assuming the solution can be expressed as the product of two functions, each depending on only one of the independent variables. It is applicable for linear and homogeneous PDEs with specified boundary conditions and simple geometry. The steps involved include rewriting the PDE, solving the separated equations, and combining the solutions. It cannot be used for non-linear PDEs, which require more advanced techniques. Other methods for solving first order PDEs include the method of characteristics, Laplace transform, and Fourier transform, depending on the specific equation and boundary conditions.
  • #1
Niles
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[SOLVED] Seperation of variables - first order PDE

Homework Statement



I have the expression X'(x)/X(x) = cx. How do I separate the variables? It's the fraction on the left side that annoys me.

I know that X'(x) = d(X(x))/dx, but I can't use this here?

EDIT: Sorry for the mis-spelled title. It's "separation".
 
Last edited:
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  • #2
The variables are X and x and they are basically already separated. The left side is (1/X)dX/dx. Move the dx to right getting d(X)/X=cxdx and integrate.
 
  • #3
Thanks.
 

1. What is separation of variables in first order PDE?

Separation of variables is a technique used to solve first order partial differential equations (PDEs). It involves assuming that the solution can be expressed as the product of two functions, each depending on only one of the independent variables in the equation.

2. When is separation of variables applicable?

Separation of variables is applicable when the PDE is linear and homogeneous, and the boundary conditions are specified for each variable separately. It is also useful when the equation has a simple geometry, such as a rectangular or cylindrical domain.

3. What are the steps involved in solving a first order PDE using separation of variables?

The first step is to rewrite the PDE in the form of a separated equation, where each variable is on one side of the equation. Then, we solve each separated equation to obtain solutions for each of the individual functions. Finally, we combine these solutions to obtain the general solution for the PDE.

4. Can separation of variables be used for non-linear PDEs?

No, separation of variables is only applicable for linear PDEs. Non-linear PDEs require more advanced techniques, such as numerical methods or the method of characteristics, to solve.

5. Is separation of variables the only method for solving first order PDEs?

No, there are other methods for solving first order PDEs, such as the method of characteristics, Laplace transform, and Fourier transform. The choice of method depends on the specific equation and boundary conditions.

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