Discretization Using ZOH, FOH and Tustis bilinear Transformation
There are a couple of steps that are required to realise a digital filter

Considering Bilinear transform 

This will facilitate taking a continuous time domain equation &amp; converting it into an appropriate discrete time domain equation. 

Copying into EE.SE just for completness

\$ H(s) = \frac{1}{1+ RCs} \$

an s-domain to z-domain transformation can be realised by substituting the s terms for the equivalent z-domain

\$ s \leftarrow \frac{2}{\Delta_T}\frac{1-z^{-1}}{1+z^{-1}}\$

\$H(z) = \frac{1+z^{-1}}{(1+ \frac{2RC}{\Delta_T}) + (1- \frac{2RC}{\Delta_T})z^{-1}}  \$

With the discrete time domain equation, the difference equation can be realised via refactoring to find the recurrence relation

\$y_n = y_{n-1} + \alpha (x_n - y_{n-1})\$

where \$\alpha = \frac{\Delta_T}{\Delta_T + \tau}\$
and \$\tau = RC\$ the time constant.

\$y_n \$ represents present output

\$y_{n-1} \$ represents last output

\$x_n\$ represents present input



If you now look at the equation (slightly re-ordered) in the question

\$u(t)=  y(t) + T\frac{dy(t)}{dt}  \$

w.r.t. the difference equation:

\$y_n = y_{n-1} + \alpha (x_n - y_{n-1})\$

It can be seen that there is a correlation. The \$\frac{dy(t)}{dt}\$ is represented by NEW - OLD and a scaling factor which has units of \$s^{-1}\$ : dy/dt



Realization in Matlab:Simulink

Looking at the difference equation: \$ y_n = y_{n-1} + \alpha (x_n - y_{n-1}) \$

there is a need for 


addition
multiplication
difference
gain 


This is using a 10kHz sin generator, sample time of 1us and an \$\alpha\$ of 0.059117397, which is a \$\tau\$ of 1.59155e-5, the equivalent of 10 kHz