# Testing Linearity of System

1. Jul 15, 2011

### mym786

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

2. Jul 15, 2011

### 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?

3. Jul 15, 2011

### HallsofIvy

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

4. Jul 16, 2011

### 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 ?

5. Jul 16, 2011

### uart

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