Find the order of the differential equation governing a system

Question

Consider a system where we are given the step response $$y(t) = 1-(1+t)e^{-t}$$ And are then asked to find the order of the differential equation which governs the system.

I have started by finding that the impulse response to the system is given by $$g(t) = te^{-t}$$ I have then found the derivative of the impulse response as $$\frac{dg}{dt}=e^{-t}-te^{-t}$$ At $t = 0$, $g(t)$ is continuous however $g'(t)$ is not (we are assuming that both are zero before $t=0$). I think this means that the system is second order but I am not sure how to reason that from the above information.

Answer 1

Not sure what exactly you are after, but note that $$y(t) = 1 - e^{-t} - te^{-t}$$ hence $$y'(t) = te^{-t}$$and so you have the obvious differential equation $$y' +y = 1-e^{-t}.$$

Can you determine the order?

Answer 2

Without some knowledge of the initial conditions we can have different differential equation that are satisfied by your equation.

The one proposed in the answer of gt6989b is one, that has solutions of the form $$ 1-(c_1+t)e^{-t} $$

that has one integration constant that can be determined by the initial condition. But we can have also the equation $$ y''+2y'+y-1=0 $$ that has solutions of the form $$ y=1-(c_1+c_2t)e^{-t} $$

with two integration constants, so, at least, we must know how many initial conditions you have.