Variations of a parameter in a differential equation

[themagiciant95]

Homework Statement

I have this differential equation:

$$a_{1}\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t)$$

and my prof, during a lesson, said that from this equation it's possible to derive that:

$$\Delta y(t)=-\Delta a_{1}\dot{y}(t)$$

Relevant Equations

(Homework Equations are already stated)

I tried to derive this by myself but I'm stuck. What i did it to substitute $a_{1}$ with $a_{1} +\Delta a_{1}$ in the first equation, getting:

$$(a_{1}+\Delta a_{1})\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t)$$

and trying to subtract $a_{1}\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t)$ to it. But it's not the right way. Can you help me ?

Ps: i think i have to make the assumption the input u(t) remains the same, right?

 

[PeroK]

I must admit, it looks a bit random to me. Are $a_1, b_0, b_1$ constants?

Are you sure it doesn't relate to the homogeneous equation?

 

[themagiciant95]

Yes, let's suppose that its a real system with the parameters $a_{1},b_{0}, b_{1}$, output y(t) and input x(t) and $\Delta a_{1},\Delta b_{0},\Delta b_{1}$ are faults the parameters

 

[PeroK]

Yes, let's suppose that its a real system with the parameters $a_{1},b_{0}, b_{1}$ and $\Delta a_{1},\Delta b_{0},\Delta b_{1}$ are faults on this parameters

In general, I can't say it makes much sense to me. There must be something in the context of the lecture.

 

[themagiciant95]

My prof also said to conjecture that the parameters change really slowly