determine if the system by theequation is invariant in time, islinear, causal and if it hasmemory?a) dy/dx + 6y(t) = 4x(t)b) dy/dx + 4ty(t) = 2×(†)

Answered: determine if the system by the equation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

determine if the system by the
equation is invariant in time, is
linear, causal and if it has
memory?
a) dy/dx + 6y(t) = 4x(t)
b) dy/dx + 4ty(t) = 2×(†)

Expert Solution

Step 1: Introduction to the problem and concepts used

Given information:

$text A. end text fraction numerator d y over denominator d x end fraction plus 6 y open parentheses t close parentheses equals 4 space x open parentheses t close parentheses text B. end text fraction numerator d y over denominator d x end fraction plus 4 space t space y open parentheses t close parentheses equals 2 space x open parentheses t close parentheses$

Aim:

To determine if the system by the equation is invariant in time, is linear, casual, and if it has memory.

To determine if a system is invariant in time, linear, causal, and has memory, we can use the following tests:

Time invariance: A system is time invariant if its output is the same for a time-shifted input. To test for time invariance, we can replace t with $t minus k$ in the system equation and see if the resulting equation is equivalent to the original equation.

Linearity: A system is linear if it satisfies the superposition principle. This means that the output of the system to a linear combination of inputs is equal to the linear combination of the outputs of the system to each individual input. To test for linearity, we can replace $x left parenthesis t right parenthesis$ with $a x left parenthesis t right parenthesis plus b x left parenthesis t right parenthesis$ in the system equation and see if the resulting equation is equivalent to $a y left parenthesis t right parenthesis plus b y left parenthesis t right parenthesis$ .

Causality: A system is causal if its output at a given time depends only on the input values up to that time. To test for causality, we can remove all future inputs from the system equation and see if the resulting equation is still solvable.

Memory: A system has memory if its output at a given time depends on the input values at previous times. To test for memory, we can remove all past inputs from the system equation and see if the resulting equation is still solvable.