Use the method of polynomial differential operators, finding the determinant and roots, and then substituting in the original first equation to eliminate extra coefficients.
Transcribed Image Text:### First-Order Differential Equations System
In this part of the lesson, we are examining a system of first-order differential equations. The system is given as follows:
\[
x' = 4x + y + 2t
\]
\[
y' = -2x + y
\]
Here:
- \( x' \) represents the first derivative of \( x \) with respect to time \( t \).
- \( y' \) represents the first derivative of \( y \) with respect to time \( t \).
- \( 4x + y + 2t \) is the expression that defines the evolution of \( x \) over time.
- \( -2x + y \) is the expression that defines the evolution of \( y \) over time.
#### Explanation:
1. The first equation \( x' = 4x + y + 2t \) indicates that the rate of change of \( x \) depends on the current value of \( x \), the current value of \( y \), and linearly on time \( t \).
2. The second equation \( y' = -2x + y \) shows that the rate of change of \( y \) depends on a linear combination of the current value of \( x \) and the current value of \( y \).
#### Analysis:
These types of equations often appear in the study of dynamical systems, control systems, and certain applications in physics and engineering, where the interactions between variables \( x \) and \( y \) need to be understood over time.
In solving such systems, various methods like the eigenvalue/eigenvector approach, numerical methods, or the method of undetermined coefficients can be applied. Understanding these equations facilitates the study of how systems evolve and predict their future behavior.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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![### First-Order Differential Equations System
In this part of the lesson, we are examining a system of first-order differential equations. The system is given as follows:
\[
x' = 4x + y + 2t
\]
\[
y' = -2x + y
\]
Here:
- \( x' \) represents the first derivative of \( x \) with respect to time \( t \).
- \( y' \) represents the first derivative of \( y \) with respect to time \( t \).
- \( 4x + y + 2t \) is the expression that defines the evolution of \( x \) over time.
- \( -2x + y \) is the expression that defines the evolution of \( y \) over time.
#### Explanation:
1. The first equation \( x' = 4x + y + 2t \) indicates that the rate of change of \( x \) depends on the current value of \( x \), the current value of \( y \), and linearly on time \( t \).
2. The second equation \( y' = -2x + y \) shows that the rate of change of \( y \) depends on a linear combination of the current value of \( x \) and the current value of \( y \).
#### Analysis:
These types of equations often appear in the study of dynamical systems, control systems, and certain applications in physics and engineering, where the interactions between variables \( x \) and \( y \) need to be understood over time.
In solving such systems, various methods like the eigenvalue/eigenvector approach, numerical methods, or the method of undetermined coefficients can be applied. Understanding these equations facilitates the study of how systems evolve and predict their future behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b9e8ebd-ceba-4fff-be05-4ffc155e08e3%2Fa9315c18-8c3d-46a6-bb33-254cee63b9c7%2F8wo6hy6_processed.png&w=3840&q=75)
