1st order ODE with singularity (and nasty particular integral)

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.

• Welcome to MSE. I am not sure right now; are you searching for a solution for the function $p_s(x)$ or for $p_v(x)$. Furthermore to what exactly are the indices $s$ and $v$ refering to? – mrtaurho Oct 5 '18 at 20:58
• Hi, thank you for replying! I am searching for $p_s(x)$, as $p_v(x)$ is known. The indices are just identifiers referring to two different pressure components, so the two p's are separte functions. – jedynygucio Oct 5 '18 at 21:01
• Ah, okay. I would encourage you to try the differential equation solver widget provided WolframAlpha which yields to a quite complicated "closed form" (technically it is not completely closed since it still contains an integral). – mrtaurho Oct 5 '18 at 21:04
• Very interesting widget. I have tried it just now, with the full form for $p_v$, but other than classifying it it does not appear to give more information. – jedynygucio Oct 5 '18 at 21:09
• I kept it simple by just setting $p_s(x)=f(x),x_T=y, p_v(x)=g$ which results as $$f(x)=\exp\left(\frac{2a\tanh^{-1}(x/y)}{y}-\log(x^2-y^2)\right)\int_0^x\frac{\exp\left(\frac{2a\tanh^{-1}(\xi/y)}{y}-\log(\xi^2-y^2)\right)(2bg+cy^2-c\zeta^2)}{\xi^2-y^2}d\xi+k_1\exp\left(\frac{2a\tanh^{-1}(x/y)}{y}-\log(x^2-y^2)\right)$$ which is quite unpleasant to look at. – mrtaurho Oct 5 '18 at 21:13

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.