Can All Differential Equations Be Separated?

[whatisreality]

I just wanted to check something. The equation

∂²φ / ∂x² + ∂²φ / ∂y² = sin(xy)

Was given as an example of a separable equation. I can't separate it, and I found online that to use separation of variables the equation should be linear, which this isn't? Is there a way of separating this?

 

[Geofleur]

Separation of variables, in this context, means writing $\phi(x,y) = X(x)Y(y)$ and getting the equation into the form (stuff depending only on $x$) = (stuff depending only on $y$) = constant. The only way this can happen is if the stuff depending only on $x$ and the stuff depending only on $y$ are are separately equal to zero. This technique, then, converts the partial different equation into two ordinary differential equations.

To use this approach, however, it's best to start with the homogeneous equation - the one without $\sin(xy)$. Once the solutions have been found for the homogeneous equation, any (reasonable) term on the right hand side can then be Fourier expanded in terms of the solutions to the homogeneous problem, and then finally the inhomogeneous equation can be solved.

 

[whatisreality]

Separation of variables, in this context, means writing $\phi(x,y) = X(x)Y(y)$ and getting the equation into the form (stuff depending only on $x$) = (stuff depending only on $y$) = constant. The only way this can happen is if the stuff depending only on $x$ and the stuff depending only on $y$ are are separately equal to zero. This technique, then, converts the partial different equation into two ordinary differential equations.

To use this approach, however, it's best to start with the homogeneous equation - the one without $\sin(xy)$. Once the solutions have been found for the homogeneous equation, any (reasonable) term on the right hand side can then be Fourier expanded in terms of the solutions to the homogeneous problem, and then finally the inhomogeneous equation can be solved.

OK, so it's possible. Is it possible without fourier? Which we haven't covered?

 

[whatisreality]

I wasn't actually told to solve it, I was just using the examples given as practice questions.

 

[Geofleur]

Well, I can't think of a way to solve it analytically (as opposed to using a computer) without using Fourier methods.

 

[whatisreality]

Well, I can't think of a way to solve it analytically (as opposed to using a computer) without using Fourier methods.

OK, thanks! Bad one to practice on :)

 

[Geofleur]

One more thing - you would also need some sort of boundary conditions (values of $\phi$ or its derivatives on the boundaries of the region of interest) to actually solve the equation, even with Fourier methods.

 

[whatisreality]

I know. I'm not trying to solve them fully, just practicing the actual separation - to the point where I have two differential equations equalling some sort of constant of separation.

 

[jedishrfu]

There are some solved examples here in video form:

https://dl.dropboxusercontent.com/u/28928849/Webpages/DifferentialEquationsVideoLibraryTable.htm

You could start the video write down the example, try to solve and then run the video to see if you did it the right way or not.

 

[whatisreality]

There are some solved examples here in video form:

https://dl.dropboxusercontent.com/u/28928849/Webpages/DifferentialEquationsVideoLibraryTable.htm

You could start the video write down the example, try to solve and then run the video to see if you did it the right way or not.

That is absolutely brilliant, thank you!

 

[differentialeq]

You can find here a lot of solved differential equations http://differential-equations.com/