This produces which is the separation of variables formula. References Paul Blanchard, Robert L. Devaney, and Glen R. Hall, Differential Equations , 4th ed. This is probably the leading text in the differential equations reform movement. I did so out of this book. Share this: Twitter Facebook. Like this: Like Loading This entry was posted in calculus , differential equations.
Bookmark the permalink. Leave a Reply Cancel reply Enter your comment here Fill in your details below or click an icon to log in:. Email required Address never made public. Name required. At this point all we want to do is identify the two ordinary differential equations that we need to solve to get a solution.
Before we do a couple of other examples we should take a second to address the fact that we made two very arbitrary seeming decisions in the above work. Both of these decisions were made to simplify the solution to the boundary value problem we got from our work. This by the way was the reason we rewrote the boundary value problem to make it a little clearer that we have in fact solved this one already. We can now at least partially answer the question of how do we know to make these decisions.
There is also, of course, a fair amount of experience that comes into play at this stage. The more experience you have in solving these the easier it often is to make these decisions. Of course, we will need to solve them in order to get a solution to the partial differential equation but that is the topic of the remaining sections in this chapter.
Once that is done we can then turn our attention to the initial condition. It just looked that way because of all the explanation that we had to put into it. First note that these boundary conditions really are homogeneous boundary conditions. If we rewrite them as,. Now, as with the heat equation the two initial conditions are here only because they need to be here for the problem. We will not actually be doing anything with them here and as mentioned previously the product solution will rarely satisfy them.
We will be dealing with those in a later section when we actually go past this first step. Again, the point of this example is only to get down to the two ordinary differential equations that separation of variables gives.
It will make solving the boundary value problem a little easier. So, after introducing the separation constant we get,. The boundary conditions in this example are identical to those from the first example and so plugging the product solution into the boundary conditions gives,.
This problem is a little well actually quite a bit in some ways different from the heat and wave equations. Also, we should point out that we have three of the boundary conditions homogeneous and one nonhomogeneous for a reason. It will often be convenient to have the boundary conditions in hand that this product solution gives before we take care of the differential equation.
Okay, now we need to decide upon a separation constant. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Is it mathematically valid to separate variables in a differential equation? Asked 6 years, 6 months ago. Active 6 years, 6 months ago. Viewed 14k times. Community Bot 1. Devarsh Ruparelia Devarsh Ruparelia 1, 1 1 gold badge 8 8 silver badges 13 13 bronze badges.
What information can provide you further context? Name of the book? The quoted statement is without context. I think the question contains more then enough information if you are familiar with how calculus is taught in the US.
If you are used to a rigorous approach to mathematical analysis it might make less sense. Show 7 more comments. Active Oldest Votes. Up until your post I only saw pages of explanations which were hard to follow :D. My introduction to PDE class in the math department has also used this technique though.
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