From the links between probability theory and analysis, we know that lots of pdf usually represent a solution to a differential equation, e.g. the normal distribution is the solution to the heat equation. However, what can be said about the Student T distribution? To my best knowledge, there is no PDE or ODE that admits as a solution the function
$$ f(x)=\frac{\Gamma \left( \frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi \sigma^2}\Gamma \left( \frac{\nu}{2}\right)} \left( 1 + \frac{(x-\mu)^2}{\nu\sigma^2} \right)^{-\frac{\nu + 1}{2}} $$
Any toughts about this question would help me a lot! :)
The pdf of the t-distribution is a solution to the following differential equation: $$ { {\begin{array}{l}\left(\nu +x^{2}\right)f'(x)+(\nu +1)xf(x)=0,\\ \\ \text{with }f(1)={\frac {\nu ^{\nu /2}(\nu +1)^{-{\frac {\nu }{2}}-{\frac {1}{2}}}}{B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)}}\end{array}}} $$ where $B(x,y)$ is a beta function.