Testing Linearity of Two Given Systems

[mym786]

Homework Statement

Two system are given as follows :-

(a) $\frac{dy}{dt}$ + sin(t)y(t) = $\frac{df}{dt} + 2f(t)$

(b) $\frac{dy}{dt}$ + 2y(t) = f(t)*$\frac{df}{dt}$

Test linearity of systems.

Homework Equations

The Attempt at a Solution

 

[uart]

So the question is basically to determine if each represents a linear or a non-linear system.

Tell us what you know or don't know. What type of things are you allowed do to the variables and their derivatives in order for it to be linear system? What are some things you can't do if you want to preserve linearity?

 

[HallsofIvy]

Main point, do you know what "linearity" means for differential equations?

 

[mym786]

My attempt to the solution.

(a) dy/dt + sin(t)y(t) = df/dt + 2f(t)

If input is f1(t) , output is y1(t).
dy1/dt + sin(t)y1(t) = df1/dt + 2f1(t) -> eqn 1

If input is f2(t) , output is y2(t)
dy2/dt + sin(t)y(t) = df2/dt + 2f2(t) -> eqn2

Now system would be linear if input is k1f1(t) and output is k1y1(t).

Let input be k1f1(t) , k2f2(t).

d(k1y1(t) + k2y2(t))/dt + sin(t)(k1y1(t) + k2y2(t)) = d(k1f1(t) + k2f2(t))/dt + (k1f1(t) + k2f2(t)). -> 3

eqn 3 is k1*eqn1 + k2*eqn2 so System is linear. The solution says System is not linear. Why ?

 

[uart]

My attempt to the solution.
...
eqn 3 is k1*eqn1 + k2*eqn2 so System is linear. The solution says System is not linear. Why ?

That's good. Your answer is correct, it is a linear system. It's not time invariant (therefore not LTI) but it is linear.