Solving Linear Differential Equations
생성일: 2024년 11월 21일
답변에 사용된 모델 OpenAI 01 Preview by OpenAI01
생성일: 2024년 11월 21일
답변에 사용된 모델 OpenAI 01 Preview by OpenAI01
Setting up the system
I’m piecing together a system of first-order linear differential equations, noting the initial conditions. This involves expressing the solutions solely in terms of real numbers.
Piecing together solution format
I’m analyzing the solution instructions, which require clarity, definition, and a structured step-by-step approach, including a heading "Subject or sub-subject" and specifics in each step.
Formulating the system
I am setting up the system in matrix form to simplify solving linear differential equations by pinpointing eigenvalues and eigenvectors.
Calculating eigenvalues
First, I calculated the determinant, set it to zero, and found the eigenvalues. Then, I computed the eigenvectors for each eigenvalue and formulated the general solution.
Mapping the journey
OK, let me see. I’m outlining the process of solving a system of first-order linear differential equations with eigenvalues and eigenvectors. This involves understanding the problem, applying formulas, and detailing the solution step-by-step.
Applying eigenvalues and eigenvectors
OK, let me see. The problem involves solving a system of first-order linear differential equations using eigenvalues and eigenvectors, with initial conditions given. This will allow us to determine functions and .
Structuring the system
I’m piecing together the system into matrix form with and to find eigenvalues and eigenvectors for the general solution.
Crunching the numbers
I’m working out the determinant of to find eigenvalues. Expanding and combining terms, it’s interesting to see the matrix's properties unfold.
Revisiting the equation
I’m working through the product . Initially, I calculated , but I’m careful to ensure accuracy with each step.
Crunching the numbers
I expanded and added 68. This gives a characteristic equation for eigenvalues.
Evaluating eigenvalues
I determined that the eigenvalues of matrix are by solving the quadratic equation derived from .
Breaking down the matrix
I’m subtracting from matrix to get the matrix , which is .
Solving for in terms of
I’m simplifying the first equation to solve for as . Substituting this into the second equation, I verify that the left side matches the right side, confirming the relationship.
Confirming the equation
Interestingly enough, establishing that aligns with the equation .
Constructing the general solution
I’m working on building the general solution by combining real and imaginary parts of eigenvectors, forming .
Adjusting my approach
I’m piecing together the general solution, ensuring and , to find .
Mapping out the solution
I'm thinking about expressing the solution for each component, considering the constants and . This involves trigonometric terms and vector constants to ensure precision.
Formulating the solution
I’m incorporating the vectors and into the general solution, , by combining constants and adjusting for initial conditions.
Weighing the options
I’m mapping out whether to use or explore and directly.
Rewriting the solution
I’m thinking through a new way to express using constants and , and providing simpler forms for and with cosine and sine functions, while noting their interdependence.
Expressing the general solution
I’m working through expressing in terms of using standard theory for systems with complex eigenvalues and eigenvectors. This involves exponential functions and trigonometric terms.
Breaking down the solution
I’m combining constants into and , then expanding component-wise to see how and combine with sines and cosines.
Breaking down the processes
I’m decomposing the expressions for and , combining exponential and trigonometric elements to streamline and group terms effectively.
Setting initial values
I determined and , finding . Now, I'll solve to pinpoint .
Breaking down the equation
First, I substitute into , arriving at . This leads me to values for and , which I'll use to formulate the solutions.
Breaking down the solutions
I derived two distinct solutions for and by substituting specific values and simplifying each term. The results included exponential decay modulated by trigonometric functions.
Subject: Solving a System of First-Order Linear Differential Equations Using Eigenvalues and Eigenvectors
Step 1: Understanding the Problem
Given:
We have the system of differential equations:
We need to find: The explicit functions and that satisfy both the differential equations and the initial conditions.
Explanation:
To solve this system, we'll use the method of eigenvalues and eigenvectors. By expressing the system in matrix form and finding the eigenvalues and eigenvectors of the coefficient matrix, we can construct the general solution.
Step-by-Step Solution:
Let . Then the system becomes:
where .
Compute the characteristic equation :
Expanding:
Solving the quadratic equation:
For , solve :
which simplifies to:
Let . From the first row:
Thus, the eigenvector is .
Step 2: Constructing the General Solution
Explanation:
With complex eigenvalues and eigenvectors, the solution involves exponential, sine, and cosine functions.
Step-by-Step Solution:
Separate into real and imaginary parts:
The general solution is:
At :
Solving:
Conclusion:
Therefore, the solutions to the differential equations are: