Question

Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system :

A. d³y/dt³ + a¹ d²y/dt²+a² dy/dt+a³y=b³u+b²du/dt+b¹ d²u/dt²
B. y(t)=∫₀ᵗeα⁽ᵗ−τ⁾βu(τ)dτ
C. y = au + b, b ≠ 0
D. y = au

Answer 1

Option B is not a linear system because the integral in the equation makes it non-linear.

Step-by-step explanation:

A linear system is a system that follows the principle of superposition and homogeneity. In a linear system, the input and output have a linear relationship. Among the given options, the system that is not linear is option B. The integral in the equation makes it a non-linear system. The integral introduces a non-linear operation, which violates the principle of homogeneity.

The question is asking which among the given systems is not a linear system. A system is considered linear if it satisfies two properties: homogeneity and additivity (superposition).

System A, described by the equation d³y/dt³ + a¹ d²y/dt²+a² dy/dt+a³y=b³u+b²du/dt+b¹ d²u/dt², is a linear differential equation assuming a and b are constants because it can be rearranged to satisfy both properties.

System B, y(t)=∫₀ᵣeα⁴ᵗ–τ)βu(τ)dτ, is linear as it represents a convolution integral with an exponential function which is also linear in terms of u.

System C, y = au + b, is not a linear system when b ≠ 0, because the presence of a non-zero constant term, b, violates the homogeneity condition of a linear system.

System D, y = au, is clearly a linear system as it satisfies both homogeneity and additivity.