Answer :

Solution :

Given : y(t)=x(e^t)

For input x1(et), output will be y1(t)

y₁(t)= x₁⁽e^t)

Multiplying both sides by a scalar quantity ' a'

ay₁(t)=ax₁(e^t)  ........(i)

For input x₂(e^t), output will be y₂(^t)

y₂(t)= x₂(e^t)

Multiplying both sides by a scalar quantity ' b'

by₂(t)= bx₂(e^t) .........(ii)

Adding equation (i) and (ii) we get

ay₁(t) + by₂(t) = ax₁(e^t) + bx₂(e^t)  .......(iii)

Let for input ax₁(e^t) + bx₂(e^t), ouput is y₃(t)

y₃(t)= ax₁(e^t) + bx₂(e^t)

But, ay₁(t) + by₂(t) = ax₁(e^t) + bx₂(e^t)

∴ y3(t)= ay1(t) + by2(t)

Hence, system satisfies both additivity and homogeneity law.

∴ The system is a linear system.

From the given system y(t)= x(et)

y(0)=x(e⁰)=x(1)

y(1)= x(e¹)=x(e)=x(2.71)

So, present value of output depends on future value of input.

∴The given system is non causal.
Hence, the correct option is (B).