So the next thing on my list is seperable differential equations. As I predicted in my proposal, I am going to have to do some work to get SymPy to recognize all equations that are separable. Currently, it can match expressions that are exactly written as 
collect(x**2*y+x**2,x**2)
but it doesn’t know yet how to separate variables in general. So I will be working on it this week. Once it can properly separate separable expressions, implementing it in dsolve will be cake.
Also, it needs to learn how to do 
Posted by Aaron Meurer 
is called homogeneous of order 
if 
for some 
out, we get 
, which is why the definition is often stated that the function can be written as a function of 
. Of course, we could do the same with 
and get a function of 
. Here is an example of a homogeneous function: 
. You can see that 
. 
and 
in the first order differential equation 
are both homogeneous functions of the same order, then the substitution 
or 
will make the equation separable. I will show how to do it for 
by the product rule. We also have 
and 
. Because these functions are homogeneous, we can “pull out” an 
from each, giving us 
. Because we made the substitution 
, we already have had to assume 
. Thus, we can divide the whole thing by 
. Refactoring the 
and 
terms, we get 
. The equation is separable! Separating variables and integrating, we get 
. If we had used 
. This is exactly how I was able to solve these equations in SymPy. Each homogeneous equation has two possible integrals, and often the right hand side of one equation is a much harder integral than the right hand side of the other. Therefore, I did them both, and applied a little heuristic on which one to return. It prefers expressions that can be solved for y (f(x) in the case of SymPy), expressions that have evaluated integrals, and if neither of them are solvable, the shortest one, which tends to be the simplest. 
. Astute readers of this blog may have noticed that this equation is 
and the equation would become seperable. If I were doing this by hand or for a homework assignment, I would make one of those substitutions and work it through, but we have the exact form above, so let’s use it just like SymPy would. 
and 
. So we get 
with 
with 
(here and for the remainder of this derivation, the arbitrary constants in the two equations do not necessarily equal each other until the last step). Both integrals can be solved with a substitution of the denominator, giving, with 
back substituted, 
and 
. If we make 
, and pull the constants in front of the logs on the right hand sides in as powers, we can combine everything into logarithms. After doing that, we then take the antilog of both sides and get 
and 
. We then raise the whole equation to the 
power and make 
. We get
is homogeneous because 
. SymPy was incapable of combining logarithms like this, so I had to write a logcombine function, which was another 100 lines of code plus tests. This came in handy, because I was also able to use it in the part above where I combined the logarithms of my answer to eliminate all of them. 
where
. If this condition holds, then there exists a function
, often called a ‘potential function’ because of some applications of theses equations to physics, such that
and
. This is because mixed partials are equal for continuous functions. The solution will then just be
. The tricky part is finding the potential function
. There is a restriction where the rectangle connecting
,
,
, and
has to lie entirely in the domain of 
and
, but if we let
, then this is not really a problem for functions that SymPy will encounter. UPDATE: It turns out this is a problem if the equation has singularities in it. Fortunately, I was able to code up a workaround that usually works.
and
. It is much simpler for a human being to do that than the above integral, because the integral involves evaluating limits and so on, but for a computer algebra system, the above integral is a one-liner.
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