Problem 11 Verify that the vector \(\mathbf... [FREE SOLUTION]

Chapter 10: Problem 11

Verify that the vector $\mathbf{X}$ is a solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=3 x-4 y \\ &\frac{d y}{d t}=4 x-7 y ; \quad \mathbf{X}=\left(\begin{array}{l} 1 \\ 2 \end{array}\right) e^{-5 t} \end{aligned} $$

Short Answer

Expert verified

$ \mathbf{X} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} e^{-5t} $ satisfies the given system.

Step by step solution

Understand the Vector and System

We are given the vector $ \mathbf{X} = \begin{pmatrix} 1 \ 2 \end{pmatrix} e^{-5t} $. This means $ x(t) = e^{-5t} $ and $ y(t) = 2 e^{-5t} $. The system of equations consists of $ \frac{dx}{dt} = 3x - 4y $ and $ \frac{dy}{dt} = 4x - 7y $. Our task is to verify that $ (x(t), y(t)) $ satisfies this system.

Differentiate the Components of $\mathbf{X}$

Calculate the derivative of $ x(t) = e^{-5t} $, which is $ \frac{dx}{dt} = -5e^{-5t} $. Similarly, differentiate $ y(t) = 2e^{-5t} $ to obtain $ \frac{dy}{dt} = -10e^{-5t} $. These represent the rate of change of $ x $ and $ y $ with respect to time.

Substitute Components into System

Substitute $ x(t) = e^{-5t} $ and $ y(t) = 2e^{-5t} $ into the left sides of the given differential equations: $ \frac{dx}{dt} = 3x - 4y $ and $ \frac{dy}{dt} = 4x - 7y $.

Calculate Right Side of First Equation

Substituting into the first equation $ 3x - 4y $, we have $ 3(e^{-5t}) - 4(2e^{-5t}) = 3e^{-5t} - 8e^{-5t} = -5e^{-5t} $, which matches $ \frac{dx}{dt} = -5e^{-5t} $.

Calculate Right Side of Second Equation

For the second equation $ 4x - 7y $, substitute to get $ 4(e^{-5t}) - 7(2e^{-5t}) = 4e^{-5t} - 14e^{-5t} = -10e^{-5t} $, consistent with $ \frac{dy}{dt} = -10e^{-5t} $.

Conclusion

Since the left side and calculated right side match for both equations, $ \mathbf{X} = \begin{pmatrix} 1 \ 2 \end{pmatrix} e^{-5t} $ is indeed a solution to the system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Solutions

In the context of differential equations, vector solutions provide a comprehensive way to represent solutions to systems of equations. A vector solution, like $ \mathbf{X} = \begin{pmatrix} 1 \ 2 \end{pmatrix} e^{-5t} $, encapsulates both components $ x(t) $ and $ y(t) $.

These components are often functions of another variable, typically time $ t $. By expressing both variables as functions within a vector, we gain an organized and compact representation of how the state of the system changes over time.

Vector $ \mathbf{X} $: Represents both state variables $ x(t) $ and $ y(t) $.
In our exercise: $ x(t) = e^{-5t} $ and $ y(t) = 2 e^{-5t} $.
Represents exponential decay due to $ e^{-5t} $.

Grasping this form is essential as it ties the system's behavior to both components simultaneously, allowing for a coordinated analysis of their evolution.

System of Equations

A system of equations in differential equations is a set of equations that define the relationships and interactions between multiple variables and their rates of change. In our exercise, the system consists of two equations:

\[\frac{dx}{dt}=3x-4y\]

\[\frac{dy}{dt}=4x-7y\] These equations connect the derivative (rate of change) of $ x $ and $ y $ to each other, portraying how changes in one affect the other.

The system defines a dynamic interaction between $ x(t) $ and $ y(t) $.
Aim is to find functions of $ t $ that satisfy both equations simultaneously.
Depicts linear relationships governed by specific coefficients.

Understanding how to set up and solve such systems is crucial for modeling more complex real-world interactions, where variables might be interdependent.

Derivatives

Derivatives are foundational to differential equations as they capture the notion of change. They represent how much a function changes as its input changes, essentially quantifying the rate of change.

In our problem, we have:

$ \frac{dx}{dt} = -5e^{-5t} $

$ \frac{dy}{dt} = -10e^{-5t} $
Both express how $ x $ and $ y $ change with respect to time.

$\frac{dx}{dt}$: Reflects change in $ x(t) $.
$\frac{dy}{dt}$: Reflects change in $ y(t) $.
Negative sign indicates decay over time.

This makes derivatives pivotal for determining the trajectory of the system over time. Grasping how to calculate and utilize derivatives is vital in solving differential systems.

Verification of Solutions

Verification is the process of ensuring that a proposed solution satisfies the given system of equations. It's a crucial step to establish confidence that the solution is correct.

In our example, after determining $ \frac{dx}{dt} $ and $ \frac{dy}{dt} $:

Substitute $ x(t) $ and $ y(t) $ back into the original equations.
Check if the calculated derivatives match the equations' right sides.

For $ 3x - 4y $ and $ 4x - 7y $, confirming left and right expressions are equal verifies the solution.

Ensures the vector $ \mathbf{X} $ fits within the system's constraints.
Vital for affirming solution accuracy.

Thorough verification dispels uncertainties and solidifies our understanding of the solution's validity.

91影视

Verify that the vector \(\mathbf{X}\) is a solution of the given system. $$ \begin{aligned} &\frac{d x}{d t}=3 x-4 y \\ &\frac{d y}{d t}=4 x-7 y ; \quad \mathbf{X}=\left(\begin{array}{l} 1 \\ 2 \end{array}\right) e^{-5 t} \end{aligned} $$

Short Answer

Step by step solution

Understand the Vector and System

Differentiate the Components of \(\mathbf{X}\)

Substitute Components into System

Calculate Right Side of First Equation

Calculate Right Side of Second Equation

Conclusion

Key Concepts

Vector Solutions

System of Equations

Derivatives

Verification of Solutions

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