It is the first time I have posted on this forum. I am a researcher working on modelling granular solid pressures in containment structures, and I have recently derived a differential equation that looks like the following:
$dp_s/dx-2(x+a)p_s/(x_T^2-x^2)=-2bxp_v(x)/(x_T^2-x^2)-c$ where $a$, $b$ and $c$ are constants.
Aside from the fact that $p_v$ is not a simple function of $x$, I am conscious that the above is a (non-?)linear ODE with a singularity in the coefficients.
I have researched a bit on the literature on 'removable' and 'essential' singularities but the writing is highly technical and I am finding it difficult to relate any of the standard results to this particular problem. Physically, the solution $p_s$ should tend to a finite limit at $x_T$, and indeed $p_s(x_T) = p_{sT}$ is the boundary condition ($p_{sT}$ is known).
I have obtained something that looks like a physical solution using a simple Euler scheme in Excel, and a numerical solution using Maple appears to support it. But I am having no luck in obtaining a closed-form solution, although one would be very welcome. According to Maple, it appears achievable, but when I compare the 'closed-form' solution with the numerical one they look nothing alike, no doubt because of the singularity.
Can I ask if anyone has worked with such equations before, and if they may know a strategy or transformation to turn it into something simpler? I am happy to give more details if required.
This is a linear DE. Writing it as $$ \dfrac{dp_v}{dx} = a(x) p_v + b(x) $$ the general solution is $$ p_v(x) = \exp(-A(x)) \left(\int \exp(A(x)) b(x)\; dx + C\right) $$ where $A(x) = \int a(x)\; dx$.
At $x = \pm x_T$ the solutions are likely to "blow up", going to $+\infty$ or $-\infty$ as you approach the singularity. What that usually corresponds to in a physical model is that something breaks.