Nonlinear time dependent equation in simulink
Seems a homework. So no full solutions will be given.

This is a simple two state variable problem. The general solution is calculated followingly:



Calc diffs is a block diagram which takes X1, X2 , possible external signals and constants as inputs, the outputs are the derivatives of the state variables. You need seemingly  two summing junctions, 3 polarity inversions, one multiplication, linear time ramp generator (=-2t) and an exponential function block for exp(-2t). 

Do not forget to set integration method, sample rate, plot scaling and the initial values to integrators.

NOTE: I have not a modern Matlab, not even a real thing, but only an ancient low cost copy. Real and new Matlab can have some things named differently.

ADDED due the comments:

You really have missed something essential about simulation. Hopefully the following diagram helps you to catch it. The integrators accumulate the time dependent inputs. 



Your system gives zero output if there are zero initial values in the integrators. I had the following simulation parameters:


time start = 0, end=2 seconds
sample rate = 10kHz (simulation step = 0.1ms)
integration = trapezoidal
initial value =1 in both integrators