a) b) d'y(t) _2d'y(1) dªy(1) _4dy(t) + 5y(t) = 6x(1) dt' di' dt¹ dt d'y(1) di' dit Property +2dy(t) _ y(t) = sin(10m)x(1 − 2) dt linear causal fixed (time invarient) dynamic system order 7 a b syste
a) b) d'y(t) _2d'y(1) dªy(1) _4dy(t) + 5y(t) = 6x(1) dt' di' dt¹ dt d'y(1) di' dit Property +2dy(t) _ y(t) = sin(10m)x(1 − 2) dt linear causal fixed (time invarient) dynamic system order 7 a b syste
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please give the reasoning for each of your answers (Y/N) for the table.
Thank you!
![**Differential Equations and System Properties**
### Differential Equations
Given the following differential equations:
a)
\[ \frac{d^3 y(t)}{dt^3} - 2\frac{d^2 y(t)}{dt^2} + \frac{d^3 y(t)}{dt^3} - 4\frac{dy(t)}{dt} + 5y(t) = 6x(t) \]
b)
\[ \frac{d^2 y(t)}{dt^2} + 2\frac{dy(t)}{dt} - y(t) = \sin(10\pi x)(t-2) \]
c)
\[ \frac{d^n y(t)}{dt^n} + \frac{d^m}{dt^m} \]
d)
\[ y \approx A(x) \widehat{=} \frac{L}{1-e^{-x(2t-1)}} \]
### System Properties Table
| Property | System |
|----------------------|------------------|
| | a | b | c | d |
| **linear** | | | | |
| **causal** | | | | |
| **fixed (time invariant)** | | | | |
| **dynamic** | | | | |
| **system order** | | | | |
(The table is given with some properties to determine for each system described by the equations a, b, c, and d. Note that system 'd' has been marked with a cross, indicating it might be excluded or incorrect.)
### Notes:
- **Linear System**: A system is linear if it adheres to the properties of additivity and homogeneity.
- **Causal System**: A system is causal if the output at any time depends only on the values of the input at the current and past times, but not future times.
- **Fixed (Time Invariant) System**: A system is time-invariant if its behavior and characteristics do not change over time.
- **Dynamic System**: A system is dynamic if its output depends not only on the current input but also on past inputs.
- **System Order**: Refers to the highest derivative of the output in the differential equation.
This educational content aims to help students](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecd9d238-92a1-4a47-af9e-c758e887edd1%2F65567c9b-87cf-4fa3-bc27-094b2e1a5c1b%2Ft408dv9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differential Equations and System Properties**
### Differential Equations
Given the following differential equations:
a)
\[ \frac{d^3 y(t)}{dt^3} - 2\frac{d^2 y(t)}{dt^2} + \frac{d^3 y(t)}{dt^3} - 4\frac{dy(t)}{dt} + 5y(t) = 6x(t) \]
b)
\[ \frac{d^2 y(t)}{dt^2} + 2\frac{dy(t)}{dt} - y(t) = \sin(10\pi x)(t-2) \]
c)
\[ \frac{d^n y(t)}{dt^n} + \frac{d^m}{dt^m} \]
d)
\[ y \approx A(x) \widehat{=} \frac{L}{1-e^{-x(2t-1)}} \]
### System Properties Table
| Property | System |
|----------------------|------------------|
| | a | b | c | d |
| **linear** | | | | |
| **causal** | | | | |
| **fixed (time invariant)** | | | | |
| **dynamic** | | | | |
| **system order** | | | | |
(The table is given with some properties to determine for each system described by the equations a, b, c, and d. Note that system 'd' has been marked with a cross, indicating it might be excluded or incorrect.)
### Notes:
- **Linear System**: A system is linear if it adheres to the properties of additivity and homogeneity.
- **Causal System**: A system is causal if the output at any time depends only on the values of the input at the current and past times, but not future times.
- **Fixed (Time Invariant) System**: A system is time-invariant if its behavior and characteristics do not change over time.
- **Dynamic System**: A system is dynamic if its output depends not only on the current input but also on past inputs.
- **System Order**: Refers to the highest derivative of the output in the differential equation.
This educational content aims to help students
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